IvIBRARY 

OF  THE 

University  of  California. 

GTF"T  OF" 


Accession  No .  6  9 J' J  C?      •    Class  No.'^     * 


Received 


THE 


FRANKLIN 


ELEMENTARY  ARITHMETIC 


EDWIN    P.  SEAVER,  A.  M. 

HEAD   MASTER   OF   THE   ENGLISH    HIGH   SCHOOL,    BOSTON  ;    FORMERLY    ASSISTANT 
PROFESSOR   OF   MATHEMATICS    IN    HARVARD   COLLEGE 


GEO.  A.  WALTON,  A.  M. 

AUTHOR  OF  WALTONS'    ARITHMETICS,    ARITHMETICAL   TABLES,    ETC. 


BOSTON 

WILLIAM    WARE    AND    COMPANY 

[Successors  to  Brewer  and  Tileston] 

1879 


6y  t>-  3c 


Copyright 
By  E.  p.  SEAVER  and  G.  A.  WALTON. 

1878. 


University  Press;  John  Wilson  &  Son, 
Cambridge. 


PREFACE. 


The  Franklin  Elementary  Arithmetic,  though  designed 
to  be  an  introduction  to  the  Franklin  Written  Arithmetic, 
is,  nevertheless,  a  complete  arithmetic  of  its  kind.  It  con- 
tains a  short  course  in  the  elements  of  numbers,  with  such 
applications  as  are  necessary  in  ordinary  business  transac- 
tions. With  the  oral  teaching  which  should  accompany 
the  use  of  any  book,  this  book  contains  enough  to  meet 
the  wants  of  that  large  class  of  pupils  who  leave  the 
schools  at  twelve  or  thirteen  years  of  age,  and  of  all  who 
desire  to  master  the  essentials  of  arithmetic  but  have  not 
time  to  study  all  that  is  found  in  the  larger  books. 

The  method  of  the  book  is  indicated  by  its  title.  It 
distinguishes  arithmetical  operations  from  the  science  of 
arithmetic.  While  it  gives  a  systematic  practice  in  the 
former,  it  leaves  the  latter  to  be  learned  after  the  mere 
operations  have  become  familiar  by  practice.  No  attempt 
is  made  to  establish  general  principles,  but  the  pupil  is 
led  to  operate  by  imitating  processes  illustrated  by  simple 
examples,  generally  in  the  concrete.  General  statements 
and  formal  rules  are  thus  rendered  unnecessary. 

The  special  features  of  the  work  are :  — 

1.  The  uniting  of  oral  exercises  with  the  written  work,  so 
that  the  same  analysis  answers  for  both  processes. 


iv  PREFACE, 

2.  The  thorough  manner  of  treating  the  four  fundamental 
operations,  especially  multiplication  and  division.  These  opera- 
tions are  so  arranged  and  combined,  that  the  oral  and  written 
processes  can  be  learned  in  the  time  ordinarily  given  to  learning 
the  multiplication  and  division  tables. 

3.  The  introduction  of  United  States  money  and  denominate 
numbers  into  the  same  sections  with  simple  numbers  and  frac- 
tions. In  this  way  the  reductions  in  United  States  money  and 
compound  numbers  are  made  a  part  of  the  necessary  practice  in 
the  fundamental  operations,  and  thus  much  time  is  saved. 

4.  Only  one  method  is  given  for  each  operation.  The  pupil 
is  thus  spared  the  perplexity  which  often  results  from  a  multi- 
plicity of  methods. 

5.  The  Appendix  to  the  book.  This  contains  full  sets  of  tables, 
with  other  methods  of  operation,  to  be  used  instead  of  those 
given  in  the  body  of  the  book,  if  the  teacher  prefers. 

6.  Drill  Tables,  by  which  the  pupil's  work  may  be  indefi- 
nitely extended,  without  the  teacher  being  required  to  search  in 
other  books  for  examples  to  apply  as  tests  in  class  exercises. 

7.  The  adaptation  of  the  exercises  for  use  in  the  class-room, 
—  all  oral  and  illustrative  examples  being  designated  by  letters 
of  the  alphabet,  while  the  examples  for  the  slate  and  blackboard 
are  numbered  consecutively  through  the  section. 

Boston,  August,  1878. 


TABLE   OF   CONTENTS. 


Reading  and  writing  Num- 
bers TO  Thousands 7 

Addition 14 

United  States  Money 20 

Subtraction 23 

Multiplication 34 

Liquid  and  Dry  Measures. . .  48 

Division 51 

Measures  of  Time 65 

Weights  and  Numbers 66 

Reading  and  writing  Num- 
bers to  Millions 72 

Factors 78 

Multiples 79 

Common  Fractions 81 

Reduction 83 

Addition 87 

Subtraction 89 

Multiplication  91 

Long  Measure 94 

Division 95 

To  find  the  whole  when  a 

part  is  given 99 


Decimal  Fractions 104 

Reduction 106 

Addition 108 

Subtraction  109 

Multiplication 110 

Division  112 

Percentage 115 

Application  to  Profit  and 
Loss,  etc 120 

Interest 122 

Bank  Discount 126 

Mensuration 130 

Area  of  Rectangles 1 30 

Square  Measure 131 

Triangles,  etc 132 

Volume     of    Rectangular 

Solids 133 

Cubic  Measure 134 

Lumber  and  Boards 135 

Prisms,  etc. 135 

Appendix 1 38 

Tables    of    Weights    and 

Measures 143 

Multiplication  Table 144 


Drill  Tables 32,  70,  76,  128 

Exercises  on  Drill  Tables 33,  71,  77,  129 

Miscellaneous  Oral  Exercises ...29,48,64,  101 

Miscellaneous  Examples  for  Slate... 29,  49,  67,  74,  90,  102,  137 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/franklinelementaOOseavrich 


fUiri7BR3 

Elementary  Arithmetic. 


SEOTIOI^    I. 

READING   AND   "V^TRITING    NUMBERS. 
Numbers  from   One  to  Ten. 


One  man. 
Two  horses. 
Three  dogs. 
Four  boys. 
Five  trees. 
Six  cows. 
Seven  sheep. 
Eight  birds. 
Nine  posts. 
Ten  rails. 


One. 

Two. 

Three, 

Four. 

Five. 

Six. 

Seven 

Eight.  8 

Nine.    9, 

Ten.    10, 


Article  1.   How  many  men  do  you  see  in  the  picture  ? 
How  many  horses  ?  boys  ?  dogs  ?  rails  ?  trees  ?  birds  ? 


8 


READING  AND    WRITING 


In  answering  these  questions,  you  have  named   some 
numbers. 

Name  the  numbers  from  one  to  ten ;  from  ten  to  one. 

2.   Copy  on  your  slate  the  figures  that  stand  for  the 
numbers  you  have  named. 


10. 


Numbers  from  Ten  to  Twenty. 

1  ^ 


CD 
<D 


17.    18.    19.     20. 


f:3 


3.    How  many  blocks  are 

a.  Ten  blocks  and  one  block  ?  11  means  ten  and  what  ? 
h.   Ten  blocks  and  two  blocks  ?     12  means  ten  and  what  ? 

c.  Ten  blocks  and  three  blocks  ?  13  means  ten  and  what  ? 

d.  Ten  blocks  and  four  blocks  ?    14  means  ten  and  what  ? 

e.  Ten  blocks  and  ^\^  blocks  ?  15  means  ten  and  what  ? 
/.  Ten  blocks  and  six  blocks  ?  16  means  ten  and  what  ? 
g>  Ten  blocks  and  seven  blocks  ?  17  means  ten  and  what  ? 
iz.  Ten  blocks  and  eight  blocks  ?  18  means  ten  and  what  ? 
i.  Ten  blocks  and  nine  blocks  ?  19  means  ten  and  what  ? 
J.    Ten  blocks  and  ten  blocks  ?      20  means  how  many  tens? 


SIMPLE  NUMBERS, 


Exercises. 

4.    Count  by  ones  from  ten  to  twenty ;  from  twenty  to  ten. 

6.    Eead  the  following : 
k.  11.  222.  18.  0.   15.  q.  19.  s.  16. 

1.    13.  n.   17.  p.   12.  r.   14.  t.   20. 

6.    Write  in  figures  on  your  slate  all  the  numbers  from 
ten  to  twenty. 


/^  //  /J  /J  /^  /S  S  ^/  Wi/f30 


Numbers  made  up  of  Tens. 
BCD 


i^qz: 


E 

"~ 

~ 

~ 

~ 

, 

20. 

30. 

40. 

50. 

100. 

Twenty. 

Thirty. 

Forty. 

Fifty. 

A  hundred. 

7.    From  the  collections  of  blocks  represented  above  find 
a.  How  many  tens  in  twenty  ?    d.  How  many  tens  in  fifty  ? 
5.  How  many  tens  in  thirty  ?     e.  How  many  tens  in  a  hun- 
c.  How  many  tens  in  forty  ? 


8.  Twenty  is  written  20. 
Thirty  is  written  30. 
Forty  is  written  40. 
Fifty  is  written      50. 


dred? 

Sixty  is  written  60. 
Seventy  is  written  70. 
Eighty  is  written  80. 
Ninety  is  written    90. 


A  hundred  is  written  100. 


10 


READING  AND   WRITING 


9.  The  figure  0  is  called  zero,  and  stands  for  no  numhei\ 

How  many  figures  are  required  to  write  any  number  of  tens 
to  nine  tens  ?    Which  figure  shows  how  many  tens  there  are  ? 

Exercises. 

10.  Count   by  tens  from   ten  to  one  hundred;    from  one 
hundred  to  ten. 

11.  Eead  the  following : 

/.   20.  h.  30.  j.   90.  L    60.  n,   10. 

g.  40.  i.   70.  k.   50.  m.   80.  o.   100. 

12.  Write  in  figures  the  numbers  made  up  of  tens  from 
ten  to  one  hundred. 

Numbers  made  up  of  Tens  and  Ones. 


y     7    y' 

i 

1 

1 

1 

1 

PL. 

1 

1 

1 

p 

m. 


13.  How  many  tens  and 
how  many  ones  are  there  in 
the  collection  marked  A?  in 

±a     H-j-Hj     y±rhi  ^  ^  i^  c  ^ 

I  t  P       M  I  I  P        H  I  I  iP        Two    tens    and    three    are 
^  ^  ^  twenty-three,  written  23. 

Three  tens  and  four  are  thirty- four,  ivritten  34. 
Four  tens  and  seven  are  forty-seven,  written  47. 
Five  tens  and  eight  are  fifty- eight,  written  58. 
Nine  tens  and  nine  are  ninety-nine,  wTitten     99. 

How  many  figures  are  needed  to  write  tens  with  ones  ? 
What  does  the  left-hand  figure  stand  for?  What  does  the 
right-hand  figure  stand  for  ? 

14.  Ones  are  called  units,  and  the  place  at  the  right 
where  the  ones  are  written  is  the  units'  place.  The  sec- 
ond place  from  the  right,  where  the  tens  are  written,  is  the 
tens'  place. 


SIMPLE  NUMBERS. 


11 


e.  From  50  to  70. 

/.   From  70  to  100. 

P' 

77. 

s. 

95. 

Q' 

84. 

t. 

46. 

r. 

69. 

u. 

81. 

Exercises. 

15.  Count  by  ones 

a.  From  20  to  30.       c.  From  30  to  50. 

b.  From  30  to  20.       d.  From  50  to  30. 

16.  Read  the  following  : 
g.   28.            J.   49.            222.  50. 
h.  35.            k.  31.            22.  63. 
2.   42.            1.   27.             o.  59. 

17.  Turn  to  page  32,  and  read  all  the  numbers  written  in  A. 

18.  E-ead  again  the  numbers  written  in  A,  and  tell  how 
many  tens  and  how  many  units  there  are  in  each  number. 
Thus,  "  Fourteen  :  one  ten  and  four  units."  "  Twenty-six  : 
two  tens  and  six  units,"  and  so  on. 

19.  Write  in  order  all  the  numbers  from  fifty  to  ninety-nine. 

Numbers  made  up  of  Hundreds. 

20.  Count  by  hundreds  from  one  hundred  to  ten  hundred. 


One  hundred  is  written  100. 
Two  hundred  is  written  200. 


Three  hundred  is  written  300. 
Nine  hundred  is  written   900. 


Exercises. 
21.    Eead  the  following : 

a.  400.  c.   700.  e.   500. 

b.  300.  d.  200.  /.   600. 


How  many  figures  are  needed  to  write  hundreds  ? 
many  of  these  figures  are  zeros  ? 


g.  900. 
h.  800. 
How 


12 


READING  AND   WRITING 


22.    Write  in  figures  the  numbers  made  up  of  hundreds 
from  one  hundred  to  nine  hundred. 


Numbers  made  up  of  Hundreds,  Tens,  and  Units. 


23.  Three  hundreds,  two  tens,  and  five 
units  are  three  hundred  twenty-five,  writ- 
ten 325. 


^       In  which  place  from  the  right  are  the 
hundreds  written  ?  the  tens  ?  the  units  ? 


Exercises. 
24.    Eead  the  following : 
i.   123.  k,   629.  m.  934.  o.  206. 

j.  304.  1.    580.  22.   468.  p.   370. 

l>Fi     J'^ivn  fn  page  32,  and  read  the  numbers  written  in  B, 

26.    Write  in  figures  the  following : 

q.    Seven  hundred  eighty-one.  t.  Kine  hundred  fifty. 

T.    Five  hundred  fifteen.  u.  Eight  hundred  six. 

s.    Three  hundred  forty-seven.  v.  Six  hundred  eleven. 

Let  the  teacher  dictate  other  numbers  made  up  of  hundreds,  tens,  and 
units  for  the  pupil  to  write. 

Numbers  made  up  of  Thousands. 

27.  Ten  hundreds  make  a  thousand, 
written  1000  or  1,000. 

Count  by  thousands  from  one  thousand 
to  ten  thousand. 

Two  thousand  is  written  2000  or  2,000. 

Three  thousand  is  written  3000  or  3,000. 

One  thousand.  -^^^  SO  on. 


SIMPLE  NUMBERS. 


13 


Exercises. 

28.    Eead  the  following : 

a.  1,000.  d.   5,000. 

b.  7,000.  e.   8,000. 

c.  3,000.  /.   6,000. 


g.  2,000. 
2z.  4,000. 
i.   9,000. 


How  many  figures  are  needed  to  write  thousands  ?     How 
many  of  these  figures  are  zeros  ? 

29.   Write  in  figures  the  numbers  made  up  of  thousands 
from  one  thousand  to  nine  thousand. 

Numbers  made  up  of  Thousands,  Hundreds,  Tens,  and  Units. 

30.  One  thousand 
three  hundred  twen- 
ty -  five  is  written 
1325. 

In  whicji  place 
from  the  right  are  the 
thousands  written  ? 

Exercises. 

31.  Read  the  following  : 

j.   1,234  m,  2,304.  p.   1,008. 

k.  5,067.  22.   5,372.  q.  3,200. 

i.    8,009.  o.  8,097.  r.    5,970. 

32.  Turn  to  page  32,  and  read  the  numbers  written  in  C. 

33.  Write  in  figures  : 

S»   One  thousand,  two  hundred. 
t.    One  thousand,  two  hundred  seventeen. 
u.  Four  thousand,  six  hundred  forty-three. 
V.  Nine  thousand,  nine  hundred  eighty. 

Let  the  teacher  dictate  other  numbers  made  up  of  thousands,  hundreds, 
tens,  and  units  for  the  pupil  to  write. 


14 


ADDITION. 


SEOTIOE"    II. 

ADDITION. 

Addition  of  I's,  2's,  and  3's  to  other  Numbers. 

^     ^^1^^^^^^  ^^'    How  many  apples 

^  are  in  the  basket  ? 

If  the  apples  in  the 
hand  and  on  the  table  be 
put  with  those  in  the  bas- 
ket, how  many  will  there 
be  in  the  basket  then  ? 

To  find  the  answer  to 
this   question,  count  the 
numbers  together  thus  :  three,  five,  six. 

36.    Counting  numbers  together  is  adding  them. 

36.  The  number  found  by  adding  is  called  the  sum, 

a.  Add  4  and  2  ;  4  and  3  ;  5  and  2  ;  5  and  3. 

b.  What  is  the  sum  of  6  and  2  ?  6  and  3  ?  7  and  2?  7  and  3  ? 

c.  What  is  the  sum  of  8  and  2  ?  8  and  3  ?  9  and  3  ?  9  and  2  ? 

Drill  Exercises. 

37.  You  w^ill  need  to  do  examples  like  the  following  over 
,       «     (.)    (5)    («    (5)    (6)     (?)    (S)    («)    andoveragain 

d.  Add  1      1     1     1     1      1     1     1      1     t^ll   yo"    can 

to     1      3      6      7      4     2      6      8      9     add  at  sight, 

—   —    —  —   —  —   —    —   —     that  is,  till  you 

e.  Add   222222222     can  name  the 

to  J    J.    J   _6    _3    _5    _9    _7    _8     sum  of  a  pair 

of  numbers  as 
/.   Add    3      3      3      3     3      3      3     3     3     ^^^^^   ^^ 

to_2JJJJJ_8_6_9     seethe  figures. 


ADDITION.  15 

38.  g.  How  many  are  2  and  1  ?  12  and  1  ?  22  and  1  ? 
42  and  1  ? 

h.  How  many  are  4  and  2  ?  14  and  2  ?  34  and  2?  44  and  2  ? 

i.  How  many  are  9  and  2  ?  19  and  2  ?  39  and  2  ?  59  and  2  ? 

j.  How  many  are  8  and  3  ?  28  and  3  ?  48  and  3  ?  68  and  3  ? 
k.  How  many  are  9  and  3  ?  19  and  3?  59  and  3?  79  and  3  ? 

39.  i.  Add  by  2's  from  2  to  20,  writing  on  your  slate  each 
sum  as  you  find  it ;  thus, 

2,  4,  6,  8,  10,  12,  14,  16,  18,  20. 

122.  Now  look  away  from  your  slate  and  add  by  2's  without 
writing  the  sums. 
In  the  same  way, 
22.    Add  by  2's  from  1  to  21. 
o.    Add  by  3's  from  3  to  30 ;  from  1  to  31 ;  from  2  to  32. 

Examples  for  the  Slate. 

40.  Copy  and  add  the  numbers  in  the  following  examples. 

In  Example  a,  begin  at  the  bottom  and  add  thus  :  "1,  3,  5,  7,  9, 
12,  15  ;  sum,  15."  Write  this  sum  under  the  line  as  it  is  written  in 
the  book.  To  see  if  your  work  is  right,  begin  at  the  top  and  add 
downwards,  thus  :  ^'3,  6,  8,  10,  12,  14,  15;  sum  again,  15." 

In  Example  b,  say  "2,  3,  4,  6,  9,  10,  13;  sum,  13  tens."  Write 
this  sum  as  it  is  Avritten  in  the  book,  putting  a  zero  to  fill  the  units' 
place.     In  the  same  way  do  the  other  examples. 


a. 

b. 

(1-) 

(2.) 

(3.) 

(4.) 

3 

30 

2 

20 

30 

300 

3 

10 

3 

30 

30 

100 

2 

30 

2 

10 

20 

300 

2 

20 

1 

30 

20 

200 

2 

10 

3 

20 

30 

100 

2 

10 

2 

10 

10 

300 

1 

20 

2 

20 

20 

300 

16 

130 

16  ADDITION. 

(5.)     (6.)  (7.)      (8.)      (9.)     ^(10.) 

3      2  300      20      300      3 

3      2  200      30      300      2 

2  1  300      30      300      3 

3  1  100      20      200      1 

1  2  300      20      200      2 
3      3  200      10      100      3 

2  3  300      30      300      3 

3  2  100                30                200                3 
1               3  200               20               200               2 

41.    In  the  next  examples,  beginning  with  the  units,  add 
the  units,  the  tens,  and  the  hundreds  separately. 

(11.)  (12.)                  (13.)                 (14.) 

Ill  221                   102                   312* 

201  101                     30                   112 

322  230                  323                  310 

203  13                   112                     33 

132  323                  232                  231 


42.  In  the  following  examples  write  the  nuriibers  to  be 
added  J  units  under  units,  tens  under  tens,  and  hundreds 
under  hundreds. 

15.  Charles  counted  201  bricks  in  one  pile,  312  in  another, 
133  in  another,  and  123  in  another.  How  many  bricks  did  he 
count  in  all  ? 

16.  Mrs.  Otis  has  four  strawberry-beds.  In  the  first  there 
are  321  plants,  in  the  second  120,  in  the  third  230,  and  in  the 
fourth  203.     How  many  plants  has  she  in  all  ? 

17.  Mr.  Otis  bought  a  horse  for  120  dollars,  another  horse 
for  230  dollars,  another  for  312  dollars,  and  a  carriage  for  330 
dollars.     How  many  dollars  did  he  pay  for  all  ? 

18.  Add  together  123,  301,  22,  210,  300,  123,  and  310. 

19.  Add  together  201,  121,  330,  12,  13,  211,  and  300. 


^ADDITION.  17 

Addition  of  4's  and  5's. 

43.    a.    If  you  have  2  cents  and  earn  4  cents  more,  how 
many  cents  will  you  have  then  ? 

b.  2  and  4  are  how  many  ?  3  and  4  ?  4  and  4  ?  4  and  3  ? 

c.  Charles  had  6  cents  and  Albert  gave  him  4  cents.     How 
many  cents  did  he  then  have  ? 

d.  How  many  are  6  and  5  ?  7  and  5  ?  7  and  4  ?  7  and  5  ? 

e.  If  you  should  make  8  snowballs  and  Dwight  should  make 
4,  how  many  would  you  both  make  ? 

/.    8  and  5  are  how  many  ?  9  and  5  ?  9  and  4  ? 

44.    Drill  Exercises. 

{1)       (^)       (3)        iU)       (5)       (G)       (?)       {8)       {9)      (10)     (11)     (IS) 

g.    Add  44444444432   3 
to   132685974879 

A.  Add  555555555232 
to   259   163748976 

45.  i.    How  many  are  4  and  4  ?  14  and  4  ?  24  and  4  ? 

J.  How  many  are  7  and  4  ?  17  and  4  ?  37  and  4  ?  47  and  4  ? 
A.  How  many  are  9  and  4  ?  29  and  4  ?  49  and  4  ?  59  and  4  ? 
1.  How  many  are  7  and  5  ?  27  and  5  ?  47  and  5  ?  67  and  5  ? 
222.  How  many  are  9  and  5  ?  19  and  5  ?  39  and  5  ?  79  and  5  ? 
22.  How  many  are  8  and  5  ?  28  and  5  ?  58  and  5  ?  98  and  5  ? 

46.  o.  Add  by  4's  from  4  to  40,  writing  down  each  sum  as  you 
find  it.     Add  by  4's  from  1  to  41 ;  from  2  to  42  ;  from  3  to  43. 

p.   Add  by  5's  from  5  to  50  ;   from  1  to  51 ;  from  2  to  52  ; 
from  3  to  53 ;  from  4  to  54. 

47.  Of.  Name  two  numbers  which  added  together  make  3. 
T.    Name  a  pair  of  numbers  which  added  together  make  4  j 

name  another  pair. 

s.    Name  a  pair  of  numbers  which  added  together  make  5 ; 
name  another  pair;  name  another  pair. 


18  ADDITION. 

Examples  for  the  Slate. 

48.  Copy  and  add  the  following : 

(20.)  (21.)           (22.)            (23.)  (24.)  (25.) 

2  2  3                 30  300  500 

3  4  2  40  500  300 
3  5  4  20  100  500 
5  4      1       50  400  200 

3  2  5       30  200  400 

4  5  2  40  500  300 
1  4               5                50      '  500  400 


49.  In  addmg  the  units  of  Example  a  you  have  12  units,  which 
you  have  learned  is  the  same  as  1  ten  and  2  units 
(Art.  13),  so  you  may  only  write  down  the  2  units, 
and  then  add  the  1  ten  with  the  tens  written  in  the 
next  column.  When  adding  the  tens  begin  with  the 
1  ten  ;  thus,  "  1,  4,  6,  8." 


a. 
425 
324 
533 


1282  In  the  same  way  do  the  following  examples. 

{2Q>.)  (27.)  (28.)  (29.)  (30.) 


301 

406 

304 

421 

330 

404 

313 

225 

312 

445 

423 

105 

514 

204 

352 

524 

432 

233 

134 

35 

32 

624 

22 

225 

110 

414 

314 

415 

542 

643 

31.  Alfred  sold  24  tomato-plants  to  one  person,  35  to  another, 
14  to  another,  42  to  another,  and  35  to  another.  How  many 
tomato-plants  did  he  sell  ? 

32.  John  picked  up  35  nuts  on  Monday,  114  Tuesday,  135 
Wednesday,  255  Thursday,  54  Friday,  and  25  Saturday.  How 
many  did  he  pick  up  in  all  ? 

33.  Add  together  204  trees,  45  trees,  543  trees,  404  trees, 
350  trees,  and  Bb  trees. 


ADDITION.  19 

Addition  of  6's  and  7's. 

60.  a.  Frank  has  set  out  3  grape-vines,  and  his  father  has 
set  out  6.     How  many  grape-vines  have  both  set  out  ? 

fe.   How  many  are  3  and  6  ?  4  and  6  ?  3  and  7  ?  4  and  7  ? 

51.  4  and  7  may  be  written  4  +  7.  The  sign  +  means 
and.     This  sign  is  read  "  plus." 

c.  How  many  are  5  +  6?  5  +  7?  Q^^Q^?  7  +  6? 

d.  Mary  knit  8  times  around  her  mitten  this  morning  and  6 
times  this  afternoon.  How  many  times  did  she  knit  around  in  all? 

e.  7  +  7  equals  what  number  ?  8  +  7  equals  what  ? 

62.   The  sign  =  means  "  equals,"  or  "  is  equal  to." 
/.    7  +  2  +  6  =  what  number  ?     6  +  3  +  7  =  what  ? 

63.    Drill  Exercises. 

(Jf)     .(f)       {3)       (A)       (5)       (6)       (7)       {8)      (9)      (10)     {11)     {12) 

g.   Add  ^^QQ>Q>Q%QQ4.b4. 
to   4215   3   9687987 

h.   Add  777777777545 
to   523164879987 

64.  i.  5  +  ^-^"^  15  +  6-?  25  +  6?  Q>-\-Q,?  76  +  6?  ^^  +  Q? 

5  +  ^?    m  +  Q,?    78  +  6? 

6  +  7?    16  +  7?    46  +  7? 

7  +  7?    27  +  7?    47  +  7? 

8  +  7?    38  +  7?   88  +  7? 
27  +  6?   85  +  7?   48  +  7? 

66.    o.   Add  by  6's  from  6  to  60,  writing  the  sums. 

p.  Add  by  6's  from  1  to  61 ;  from  2  to  62 ;  from  3  to  63 ; 
from  4  to  64 ;  from  5  to  ^6. 

q.  Add  by  7's  from  7  to  70 ;  from  1  to  71 ;  from  2  to  72 ; 
from  3  to  73 ;  from  4  to  74;  from  5  to  75;  from  6  to  76. 

^Q.  r.  Name  each  pair  of  numbers  which  added  together 
make  6 ;  which  make  7 ;   8  ;   9 ;    10. 


J- 

7  +  6-? 

17  +  6=? 

37  +  6? 

k. 

9  +  6  =  ? 

29  +  6  =  ? 

49  +  6? 

1. 

4  +  7-? 

34  +  7  =  ? 

54+7? 

m. 

,5+7=? 

45  +  7  =  ? 

75  +  7? 

22. 

9  +  7  =  ? 

19  +  7  =  ? 

69  +  7? 

20 


ADDITION, 


UNITED    STATES    MONEY. 


67.    Ten  cents  make  a  dime. 
Ten  dimes  make  a  dollar. 

One  hundred  cents  make  a  dollar. 
The  sign  $  stands  for  dollars.    The  sign  f  stands  for  cents. 


68.    Examples  for  the  Slate. 

What  is  the  sum  of  $37,  $502,  $657,  $462, 


102 


34. 
$735? 

35.   What  is  the  sum  of  35/,  72/,  45/,  9/, 
and  47/? 

69.    Dollars   and    cents    are   written    together, 
twenty-five  dollars  and  thirty-six  cents  are  written 


(36.) 
$707 
674 
547 
436 
776 


(37.) 
$1.67 
2.34 
7.06 
6.57 
7.65 


(38.) 
$7.12 
4.37 
3.24 
2.16 
5.74 


(39.) 
$  12.42 
17.32 
62.45 
77.23 
54.36 


\1^,  and 
^  604/, 

Thus, 
$25.36. 
(40.) 

$  54.07 
67.35 
44.72 
52.47 
77.64 


60.  In  order  to  add  the  numbers  in  the  following  example 
easily,  write  them  dollars  under  dollars  and  cents  under  cents. 

41.  M}^  sister  went  shopping,  and  bought  a  veil  for  72  /,  a  pair 
of  gloves  for  $1.25,  some  slippers  for  $2.25,  some  cambric  for 
$  3.57,  and  had  $  7.65  left.    How  much  money  had  she  at  first  ? 


ADDITION,  21 

Addition  of  8's  and  9's. 

61.   a.   Ira  has  2  school-books  and  James  has  8.    How  many 
school-books  have  both  together  ? 

b.   How  many  are  2  +  8  ?   3  +  8?   3  +  9?   2  +  9? 
-  c.   Ira  made  4  paper  bags  to-day  and  James  made  8.     How 
many  paper  bags  did  both  make  ? 

d.  How  many  are  4  +  8  ?    4  +  9?    5  +  8?    5  +  9? 

e.  If  you  have  6  pencils  and  buy  8  more,  how  many  pencils 
will  you  then  have  ? 

/.   Add  together  7  pencils  and  8  pencils ;  6  pencils  and  9 
pencils. 

^.  2  +  5  +  9  =  what?    2  +  6  +  8?    5  +  3  +  9?    5  +  4  +  9? 

62.    Drill  Exercises. 

(1)      {2)       (5)       (A)       (5)       (6)       (7)       (5)       (9)      (10)     {11)     {IS) 

h.   Add  888888888676 
to  685231479768 


i.    Add   9     9      9 

9      9 

9     9 

9     9 

7     6     7 

to     3     6     2 

1      5 

4     8 

7     9 

8      9      7 

63.  j.  4  +  8?   14  + 

8?   34  + 

8?    6  + 

8?    26  + 

8?    66  +  8? 

k.   5  +  8?    25  +  8? 

45  +  8? 

8  +  8? 

38  +  8? 

68  +  8? 

1.    9  +  8?    39  +  8? 

69  +  8? 

7  +  8? 

17  +  8? 

27  +  8? 

m.  4  +  9?    44  +  9? 

94  +  9? 

6  +  9? 

16  +  9? 

36  +  9? 

22.  5  +  9?    25  +  9? 

46  +  9? 

7  +  9? 

27  +  9? 

57  +  9? 

0.  8  +  9?    18  +  9? 

48  +  9? 

9  +  9? 

29  +  9? 

99  +  9? 

64.  p.  Add  by  8's  from  8  to  80 ;  from  1  to  81 ;  from  2  to 
82 ;  from  3  to  83 ;  from  4  to  84 ;  from  5  to  85 ;  from  6  to  86; 
from  7  to  87. 

q.  Add  by  9's  from  9  to  90;  from  1  to  91;  from  2  to  92 ; 
from  3  to  93 ;  from  4  to  94 ;  from  5  to  95 ;  from  6  to  96 ; 
from  7  to  97 ;  from  8  to  98. 

65.  r.  Name  each  pair  of  numbers  which  added  together 
make  11;  12;  13;  14;  15;  16;  17;  18;  19. 


22  ADDITION. 

QQ»    Examples  for  the  Slate. 
Copy  and  add  the  following : 


(42.) 

(43.) 

(44.) 

(46.) 

(46.) 

$3.28 

$9.19 

$0.36 

$9.64 

$8.19 

32.35 

8.20 

7.21 

9.65 

7.28 

5.92 

7.39 

8.92 

8.70 

6.37 

8.67 

6.48 

6.79 

2.78 

5.00 

7.98 

5.57 

5.43 

7.87 

6.67 

0.83 

4.66 

2.19 

9.86 

7.68 

6.77 

3.75 

0.87 

6.93 

8.79 

0.06 

2.84 

6.54 

2.39 

9.80 

47.  I  bought  a  hat  for  13.75,  a  coat  for  $12,  a  pah-  of 
gloves  for  §  1.75,  some  handkerchiefs  for  $  3,  and  a  pair  of 
hoots  for  $  7.50.     What  was  the  cost  of  all  ? 

48.  How  much  money  must  you  have  that  you  may  buy  a 
dictionary  for  $1.25,  a  slate  for  35,^,  a  ruler  for  12/,  a  geog- 
raphy for  $  1.37,  an  arithmetic  for  92/,  a  reader  for  87/,  and 
a  writing-desk  for  $  2.35  ? 

49.  My  father  owes  $25.78  to  one  man,  $79.48  to  another, 
97/  to  a  third,  $75.94  to  a  fourth,  and  $5.48  to  me.  How 
much  money  does  he  owe  in  all  ? 

50.  Charles  bought,  at  a  grocery-store,  a  barrel  of  flour  for 
$9.35,  butter  for  $17.36,  beans  for  $3.50,  fish  for  92/,  eggs 
for  58/,  and  $  18.15  worth  of  sugar.    What  did  tlie  whole  cost? 

51.  Mr.  Smith  spent  during  the  winter  months  $  18.88  for 
groceries,  $  15.91  for  meat,  $  20  for  fuel,  $  36  for  rent,  and  $  84 
for  clothing.     What  did  he  spend  in  all  ? 

62.  What  is  the  sum  of  $  5.79,  $87.68,  $9.87,  $87.38,  $17, 
and  $  4.82  ? 

53.  What  is  the  sum  of  $4.35,  $25.98,  $8.37, 13/,  $38.49, 
and  $1.67? 

54.  Add  $9.98,  $17.74,  $0.49,  $15,  $23.05,  and  6Q^f  ? 

For  other  examples  in  Addition,  see  page  33. 


SUBTRACTION, 


23 


SECTION    III. 

SUBTRACTION. 
Subtraction  of  I's,  2's,'and  3's  from  other  Numbers. 

67.  What  is  the  whole 
number  of  apples  in  the 
basket?  If  the  two  apples 
under  the  hand  be  taken 
away,  how  many  will  be 
left  in  the  basket  ? 
68.  Taking  part  of  a  num- 
ber away  to  find  how  many 
are  left  is  subtracting. 

69.  The  number,  part  of  which  is  to  be  taken  away,  is 
the  minuend. 

70.  The  part  of  the  minuend  to  be  taken  away  is  the 
subtrahend. 

71.  The  part  of  the  minuend  left  after  a  part  has  been 
taken  away  is  the  remainder. 

a.  Subtract  2  from  5  ;  3  from  5  ;  2  from  6  ;  3  from  6. 

b.  If  you  subtract  2  from  1,  how  many  will  be  left  ? 

c.  What  will  be  the  remainder  if  3  be  subtracted  from  4  ? 
from  7  ?  from  8  ?  from  11  ?  from  12  ? 

Drill  Exercises. 

72.  You  will  need  to  do  examples  like  the  following 
until  you  can  subtract  at  sight. 

(7)    (8)    (9)     m 

8      7      9      10 
1111 


{!) 

(^) 

{3) 

ih) 

(-5) 

(6) 

d.  Prom       2 

3 

6 

5 

1 

4 

subtract  1 

1 

1 

1 

1 

1 

24 


I: 

SUBTRACTION. 

(1) 

(2) 

(S) 

(-4) 

(5)         (6) 

(7) 

(5) 

(i?) 

(iO) 

e.  From       4 

11 

5 

3 

8       6 

9 

7 

2 

10 

subtract  2 

2 

2 

2 

2       2 

2 

2 

2 

2 

/.    From       4 

3 

T 

5 

10     12 

8 

11 

6 

9 

subtract  3 

3 

3 

3 

3       3 

3 

3 

3 

3 

78.    g.   How  many  are  11  less  2  ?  21  less  2  ?  31  less  2  ? 
A.  11  less  3  ?  21  less  3  ?  51  less  3  ?  71  less  3  ?  91  less  3  ? 
i.    12  less  3  ?  42  less  3  ?  62  less  3  ?  82  less  3  ?  92  less  3  ? 

74.  j.    Subtract  from  20  by  2's,  writing  down  each  remain- 
der as  you  find  it ;  thus,  20,  18,  16,  14,  12,  and  so  on. 

k.  Now  subtract  from  20  by  2's  without  writing  the  remain- 
ders.    Subtract  in  the  same  way  by  2's  from  21. 
1,    Subtract  by  3's  from  30 ;   from  31 ;   from  32. 

Examples  for  the  Slate. 

75.  Copy  the  following  examples  upon  your  slate  and 
subtract,  taking  units  from  units,  tens  from  tens,  hundreds 

from  hundreds, 
and  writing  the 
remainders  as 
shown  in  Ex- 
amples a  and  h- 
To  see  if  your  work  is  right,  Add  the  remainder  to  the 
subtrahend.     The  suni  ought  to  equal  the  mimiend. 

76.  In  Example  c  the  2  tens  are  taken  from  11  tens. 
Subtract  in  the  same  way  in  the  other  examples. 


a. 

b. 

(1-) 

(2.) 

(3.) 

(4.) 

33 

43 

56 

125 

468 

359 

12 

33 

20 

13 

121 

202 

21 

10 

c.               (5.)              (6.)              (7.) 

(8.) 

(9.) 

L15              114              410              205 

911 

328 

23               21              102                31 

103 

32 

92 

10.   If  you  take  32  from  127,  how  man 

y  will  be  left? 

11.    How  many  are  105  nuts  less  22  nuts  ?    112  less  31  ? 


SUBTRACTION.  25 

Subtraction  of  4's,  5's,  and  6's. 

77.  a.   Alvin  had  5  melons  and  sold  4  of  them.     How  many 
melons  had  he  left  ? 

b.  How  many  are  6  less  4  ?    7  less  4  ?    7  less  5  ?    7  less  6  ? 

78.  7  less  6  may  be  written  7-6.     The  sign -means 
less.     This  sign  is  read  "  minus  "  or  "  less." 

c.  How  many  are  8  -  6  ?   8-4?   8-5?   9-5? 

d.  Jane  found  9  ripe  peaches  and  Mary  found  4.     How 
many  more  peaches  did  Jane  find  than  Mary  ? 

e.  How  many  more  are  10  than  4  ?   than  5  ?   than  6  ? 

/.    If  you  have  11  cents  and  spend  5  of  them,  how  many 
cents  will  you  have  left  ? 

g.    If  you  have  11  and  spend  6^  how  many  will  you  have  left  ? 
h,  12 -4  are  how  many?  12-5?  12-6?  13-6?  13-5? 


79.    DriU  Exercises. 

(1) 

(3)       (3)        (4)        (5)         (6) 

(7) 

(8) 

(9) 

(10 

i. 

From       13 

5      8      4      7      11 

9 

10 

6 

12 

subtract    4 

4.      A      4.      4c        4. 

4 

4 

4 

4 

jf.   From        5 

14 

7 

8 

11 

6 

9 

13 

10 

12 

subtract    5 

5 

6 

5 

5 

5 

5 

5 

5 

5 

k.   From       10 

8 

11 

14 

7 

13 

12 

15 

6 

9 

subtract     6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

80.  i.    How  many  more  are  8  than  4?   18  than  4?    11 
than  4  ?   21  than  4  ? 

221.  How  many  are  12  -  4  ?    22  -  4  ?    13  -  4  ?    53  -  4  ? 
n.  11-5?    21-5?    41-5?    12-5?    32-5?    42-5? 
0.   13-5?    23-5?    53-5?    14-5?    44-5?    64-5? 
p.   11-6?    21-6?    31-6?    13-6?    23-6?    43-6? 

81.  q.  Subtract  by  4's  from  40 ;  from  41 ;  from  42 ;  from  43. 
r.    Subtract  by  5's  from  50  ;  from  51 ;  from  52  ;  from  53. 
s.  Subtract  by  6's  from  60 ;  from  61 ;  from  62  ;  from  63. 


26  SUBTRACTION. 

82.    Examples  for  the  Slate. 

(12.)          (13.)          (14.)          (15.)  (16.)  (17.) 

From       247            125  ♦         712           513  138  714 

subtract  134              43            405           306  54  206 

83.    In  Example  a,  as  we  cannot  take  4  from  2,  we  change  one  of 
the  8  tens  (leaving  7  tens)  to  units.      This  1  ten  equals 
(7){i2)       10  units.      Ten  units  and  2  units  are  12  units.     We  now 
8  2       take  4  units  away  from  12  units  and  have  8  units  left. 
2  4  Next  we  take  2  tens  from  7  tens,  and  have  5  tens  left, 

r  o       and  we  have  for  the  whole  remainder  5  tens  and  8  units, 
or  58. 
In  subtracting,  say  "  4  from  12,  8 ;  2  from  7,  5.     Ans.  58." 

In  the  same  way  you  may  do  the  following  examples. 

(18.)  (19.)  (20.)  (21.)  {22.)  (23.) 

262  374  560  305  618  736 

125  236  414  154  432  144 


(24.) 

(26.) 

(26.) 

(27.) 

(28.) 

1 21.40 

$  33.62 

$42.34 

180.91 

1 73.85 

15.36 

26.16 

14.05 

25.44 

65.16 

29.  A  train  of  cars  took  425  passengers  into  Buffalo  and 
brought  back  only  216.  How  many  more  passengers  were 
taken  into  Buffalo  than  were  brought  back? 

30.  In  the  Bates  School  are  527  pupils  and  in  the  Lincoln 
School  are  455  pupils.  How  many  more  pupils  are  in  the 
Bates  School  than  in  the  Lincoln  ? 

31.  In  a  certain  town  are  642  children,  of  whom  only  526 
attend  school.     How  many  do  not  attend  school  ? 

32.  Mrs.  Eay  bought  a  house  for  $2450,  but  paid  only 
$1625  down.     How  much  does  she  still  owe? 

33.  James's  father  made  625  pounds  of  maple  sugar  last 
year,  and  has  made  1242  pounds  this  year.  How  many  more 
pounds  has  he  made  this  year  than  he  made  last  ? 


SUBTRACTION. 


Subtraction  of  7's,  8's,  and  9's. 


84.    a.   Ned's  father  had  10  horses  and  sold  7  of  them. 
How  many  horses  had  he  left  ? 

b.  Seven  from  10  leaves  what  ?  8  from  10  ?  9  from  10  ? 

c.  Mary  is  11  years  old  and  Ethel  is  6.     What  is  the  dif- 
ference of  their  ages  ? 

d.  Howmanyarell-8?  11-9?  12-9?  12-7?  12-8? 

e.  John  lives  13  miles  from  his  cousin.    After  going  7  miles 
on  the  way,  how  many  more  miles  has  he  to  go  ? 

/.    What  number  added  to  7  will  make  13  ?  14  ?  15  ? 
g.  What  number  added  to  8  will  make  13  ?  14  ?  15  ? 
h.  Emma  had  16  rabbits  and  gave  away  all  but  7.     How 
many  did  she  give  away  ? 

i.    What  number  taken  from  16  will  leave  7  ?  8  ?  9  ? 

85.    Drill  Exercises. 

{1)  {2)        (3)       iU)       (5)         (6')  (7)  {8)         (9)         {10) 

j.   From       12      11      7      9      8      10      13      15      16      14 

subtract    7        77-77        7        7        7        7        7 


k.  From       11 

13 

10 

8 

12 

9 

15 

14 

17 

16 

subtract    8 

8 

8 

8 

8 

8 

8 

8 

8 

8 

1.   From       12 

17 

15 

10 

11 

9 

13 

16 

14 

18 

subtract    9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

86.  m.  11  -  7  ?  21  -  7  ?  41  -  7  ?  12  -  7  ?  62  -  7  ?  72  -  7  ? 
n,  14-7?  24-7?  54-7?  16-7?  46-7?  96-7? 
0.  13-7?  43-7?  83-7?  15-7?  25-7?  75-7? 
p.  11-8?  31-8?  51-8?  13-8?  23-8?  53-8? 
q.  15-8?  25-8?  45-8?  17-8?  37-8?  87-8? 
r.  14-8?  44-8?  84-8?  13-8?  33-8?  73-8? 
-s-.  13-9?  33-9?  63-9?  15-9?  25-9?  85-9? 
t.    17-9?  67-9?     87-9?  16-9?    46-9?  96-9? 

87.  u.  Subtract  by  7^8  from  70;  from  71;  from  72;  from 
73;  from  74;  from  75;  from  76. 


28  SUBTRACTION. 

V.  Subtract  by  8's  from  80 ;  from  81 ;  from  82  ;  from  83  ; 
from  84 ;  from  85 ;  from  86 ;  from  87. 

w.  Subtract  by  9's  from  90 ;  from  91 ;  from  92 ;  from  93  j 
from  94 ;  from  95 ;  from  96 ;  from  97. 

Examples  for  the  Slate. 

88.    a.    Subtract  168  from  300. 

WRITTEN  WORK.         In  this  example,  as  we  have  no  units  to  take  the 
(2)  (9)  iio)  8  units  from,  and  no  tens  to  take  the  6  tens  from, 

3  0  0  we  change  one  of  the  hundreds  (leaving  2  hundreds) 

16  8  to  10  tens,  and  then  change  one  of  the  tens  (leaving 

13  2  ^  tens)  to  ten  units.     We  have  now  2  hundreds, 

9  tens,  and  10  units,  from  which  we  can  easily  take 
the  1  hundred,  6  tens,  and  8  units. 

In  subtracting,  say  "  8  from  10,  2  ;  6  from  9,  3  ;  1  from  2,  1." 


(34.) 

(35.) 

(36.) 

(37.) 

(38.) 

(39.) 

400 

502 

800 

$  17.28 

1 98.76 

$86.42 

285 

259 

769 

5.79 

67.89 

35.69 

40.  Mary  took  $5.10  to  the  store,  and  bought  a  pair  of 
boots  for  $2.38.     How  much  money  had  she  left? 

41.  A  man  bought  a  horse  for  $  378  and  sold  him  for  $  417. 
How  many  dollars  did  he  gain  ?  • 

42.  Sarah's  father  bought  her  a  piano  for  $425,  paying  all 
but  $287  of  the  money  at  the  time  he  bought  it.  How  much 
did  he  pay  ? 

43.  A  house  and  lot  together  were  worth  $4500,  and  the 
land  alone  was  worth  $  1185.     What  was  the  house  worth  ? 

44.  Otis  bought  of  Mr.  Carter  cloth  to  the  amount  of  $  29.80, 
and  Mr.  Carter  took  off  $  2.98  of  the  cost,  for  present  payment. 
How  much  did  Otis  then  have  to  pay  ? 

45.  Mr.  Diller  raised  1280  baskets  of  peaches  this  year  and 
927  baskets  last  year.  How  many  more  baskets  of  peaches 
did  he  raise  this  year  than  last  ? 

For  other  examples  in  Subtraction,  see  page  33. 


MISCELLANEOUS  EXAMPLES,  29 

89.    Miscellaneous  Oral  Examples. 

a.  James  has  24  doves,  and  Walter  has  9  more  than  James. 
How  many  doves  has  Walter  ? 

b.  If  Burt  has  26  chestnuts  and  gives  9  of  them  to  Ida  and 
7  to  you,  how  many  will  he  have  left  ? 

c.  Annie  had  50/,  and  bought  with  it  some  paper  for  9/, 
some  ink  for  10/,  and  an  orange  for  6  ^.  How  many  cents  had 
she  left  ? 

d.  Add  9  to  32  and  then  subtract  7  from  the  sum. 

e.  From  59  take  the  sum  of  14  and  6. 

/.  A  boy  threw  a  ball  7  feet  beyond  the  fence.  If  the  fence 
was  24  feet  from  the  boy,  how  far  must  he  go  to  get  the  ball  ? 
How  far  to  get  the  ball  and  return  ? 

g.  If  John  is  7  feet  from  you  on  the  right  and  Alice  14  feet 
from  you  on  the  left,  how  far  is  John  from  Alice  ?  How  far 
must  John  walk  to  go  past  you  to  Alice  and  then  back  to 
you  again? 

h.  The  end  of  the  school-room  is  30  feet  long.  There  are  2 
doors  in  the  end  each  4  feet  wide,  and  the  rest  of  the  distance 
is  filled  with  blackboards.  How  many  feet  are  filled  with 
blackboards  ? 

i.  Another  school-room  is  33  feet  long.  Ellen  is  standing 
at  one  end,  and  Jane  is  standing  opposite  her  at  the  other  end. 
If  Jane  now  walks  5  feet  towards  Ellen,  and  Ellen  walks  7  feet 
towards  Jane,  how  far  apart  will  they  be  ? 

90.    Miscellaneous  Examples  for  the  Slate. 

46.  If  a  boy  earns  $  9.50  and  spends  $  2.35  for  books,  how 
much  money  will  he  have  left  ? 

47.  Grace  has  1 29.63  and  Charlotte  has  $  7.85  more  than 
Grace.     How  much  money  has  Charlotte  ? 

48.  Mr.  Rolfe  owned  435  acres  of  land  and  sold  42  acres  to 
one  person  and  37  acres  to  another.  How  many  acres  had  he  left ? 

49.  The  material  for  Kate's  dress  cost  $5.84,  the  making 
cost  $  3.15  and  the  trimming  $  1.75.  What  did  the  whole  dress 
cost? 


30  SUBTRACTION. 

50.  A  bookcase  has  5  shelves.  On  the  first  are  24  books ; 
on  the  second,  37 ;  on  the  third,  45 ;  on  the  fourth,  48 ;  and 
on  the  fifth,  52.  If  4  books  be  taken  from  the  first  shelf,  8 
from  the  second,  and  17  from  the  third,  how  many  books  will 
remain  in  the  case  ? 

51.  Mrs.  Bush  lent  a  neighbor  $  20.  He  paid  her  at  one 
time  $  6.50,  at  another  time  I  b.S^,  and  at  another  time  gave 
towards  jDayment  of  the  debt,  a  calf  worth  $  7.50.  How  much 
of  the  debt  did  he  then  owe  ? 

52.  Mr.  Mason  paid  $  121.80  for  his  passage  to  Liverpool, 
and  $  26.32  more  for  his  jDassage  back  than  for  his  passage  out. 
How  much  did  he  pay  for  his  passage  back  ? 

53.  A  grocer  paid  $  18  for  a  barrel  of  sugar  and  $  0.75  for 
freight  upon  it.  For  how  much  must  he  sell  it  to  gain 
$6.75? 

54.  Mr.  Gove's  income  this  year  is  $  568  besides  his  salary 
of  $  1500.  He  pays  $  250  for  rent,  $  67  for  fuel,  and  a  tax  of 
$  48.     How  much  has  he  left  for  his  other  expenses  ? 

^^.  Dora's  mother  gave  her  a  5-dollar  bill,  with  which  she 
bought  some  cambric  for  $1.55,  some  thread  for  10/,  buttons 
for  37^,  and  a  sack  pattern  for  35/.  How  much  change  should 
she  receive  in  return  ? 

b^.  Find  the  sum  of  $  3684  and  $  2748,  and  then  subtract 
from  this  sum  $  197. 

57.    From  $  30.07  take  the  sum  of  $  5.87  and  62/. 

bS.  A  man  who  has  $  8600  owes  $  2107  to  one  person  and 
$  3648  to  another.  What  will  he  have  left  after  paying  what 
he  owes  ? 

59.  A  man  owned  a  house  that  cost  $  5285  and  a  barn  that 
cost  $  670 ;  he  was  obliged  to  sell  them  both  for  $  4500.  How 
much  money  did  he  lose  by  the  sale  ? 

60.  John  has  264/,  which  is  187/  less  than  Charles  has. 
How  many  cents  has  Charles  ? 

61.  Find  the  difference  between  284  and  109,  and  then  take 
that  difference  from  248  +  2964  4-  307. 


MISCELLANEOUS  EXAMPLES. 


31 


62.  A  man  bought  a  carriage  for  $465,  spent  $38  for  re- 
pairing it,  and  then  sold  it  for  $  485.  Did  he  gain  or  lose,  and 
how  mnch  ? 

63.  James  shot  an  arrow  74  feet  and  Fred  shot  one  2S  feet 
farther  than  James.  How  many  feet  must  Fred  go  to  pick 
up  his  arrow  and  return  ? 

64.  Ralph  lives  150  yards  north  of  the  school-house.  One 
morning  he  went  for  Arthur,  who  lives  85  yards  south  of  the 
school-house,  and  then  went  to  school.  How  many  yards  did 
he  travel  ? 

Q5.  Two  girls  start  from  the  same  place  to  roll  hoops.  One 
goes  east  298  feet,  and  the  other  west  284  feet.  How  far 
apart  are  they  then  ? 

QQ>.  If  the  two  girls  are  582  feet  apart  and  should  go  towards 
each  other,  one  132  feet  and  the  other  175  feet,  how  far  apart 
would  they  he  then  ? 

67.  From  a  tank  containing  412  gallons  of  water  148  gal- 
lons were  drawn  off,  after  which  327  gallons  ran  in.  How 
many  gallons  were  there  in  the  tank  then  ? 

^S.  Here  is  a  boy's  account  of  what  he  received  and  w^hat 
he  spent  last  June.  How  much  has  he  left  to  begin  his  July 
account  with  ? 


June  1 


17 


June  3 
"  6 
"  8 
"  15 


On  hand  from  last  month 

Received  from  father 

For  doing  an  errand 

For  weeding  for  Mrs.  Bates,  4  hours . 
Mother  gave  me 


Amount. 


Spent  for  cap,  75 f;  for  slate,  S7  f 

Qave  Mrs.  Fox  JfOf ;  bought  ball,  35 f 

Paid  for  square  of  glass,  J^Of ;  for  paper,  5f\ 

Birthday's  present  for  Susie 

Bought  shoes,  ^  2.25  ;  hoe,  50  f 


Amount 

Balance  on  hand  June  30,  B 


42 
00 
10 
50 
25 


25 


32 


DRILL    TABLE. 


92.    DRILL   TABLE   No.  1. 
Simple  Numbers. 


A 

B 

c 

D 

E 

F 

gh 

ijk 

I    mno 

P'l 

rs  t 

u    vwx 

14 

752 

4  505 

25 

324 

1  231 

26 

720 

5  604 

48 

747 

7  038 

46 

237 

6  162 

37 

421 

5  370 

32 

471 

7  390 

16 

908 

6  325 

26 

623 

8  919 

84 

162 

2  591 

45 

305 

7  052 

98 

143 

4  028 

21 

750 

7  644 

32 

681 

3  237 

33 

626 

5  381 

18 

645 

9  804 

42 

638 

2  486 

47 

970 

8  275 

45 

242 

1  827 

54 

352 

5  129 

31 

327 

9  413 

81 

147 

2  624 

34 

428 

3  796 

72 

638 

4  140 

15 

514 

4  468 

45 

902 

1  561 

23 

762 

8  574 

32 

540 

7  018 

35 

644 

7  631 

29 

412 

5  381 

44 

371 

3  628 

17 

915 

4  973 

41 

427 

2  376 

41 

109 

9  462 

36 

755 

6  705 

95 

484 

4  092 

45 

237 

8  892 

82 

723 

8  407 

24 

642 

4  534 

34 

528 

1  295 

32 

768 

2  347 

66 

892 

3  Oil 

45 

374 

3  926 

19 

717 

2  814 

42 

461 

9  319 

78 

158 

5  390 

26 

175 

9  235 

39 

243 

4  321 

15 

780 

1  527 

46 

905 

j  6  804 

DRILL  EXERCISES.  \  \\ 

93.     Explanation  of  the  Use  of  the  Drill  Tables. 

The  following  illustration  will  show  how  the  table  on 
page  32  may  be  used  for  class  drill  so  as  to  give  each  pupil 
a  different  example : 

1.  Let  the  members  of  the  class  number  themselves  1,  2,  3, 
etc.,  to  any  given  number  up  to  25 ;  and  let  each  member  find 
his  number  in  the  left-hand  margin  of  the  table. 

2.  The  teacher  then  gives  a  direction  in  this  form :  "  Add 
A,  D,  and  G." 

3.  In  obedience  to  this  direction,  each  pupil  will  add  the 
numbers  that  he  finds  expressed  under  the  letters  A,  D,  and  Gr, 
and  in  the  line  of  his  own  number.  Thus,  pupil  No.  1  will  add 
14,  25,  and  23 ;  No.  2  will  add  26,  48,  and  38 ;  and  so  on. 

Thus  a  series  of  examples  is  given  out  at  a  single  dictation, 
and  the  pupils  are  taught  to  work  independently. 

The  key  contains  answers  to  all  these  examples. 
94.     Exercises  on  Table  No.  1. 


Addition. 

1.  Add  A,  D,  and  G. 

2.  Add  A,  B,  and  C. 

3.  Add  D,  E,  F,  and  G. 

4.  Add  B,  C,  D,  and  E. 

5.  Add  A,  B,  0,  D,  E,  and  269. 

Subtraction. 

6.  From  B  take  A. 

7.  From  E  take  D. 

8.  From  C  take  B. 

9.  Subtract  E  from  F. 

10.   Find  the  difference  between  C 
andF. 


Oral  Practice. 

11.  Add  g  and  h ;  add  j;  and  q ;  add 

y  and  z. 

12.  Add  %y  j,  and  Tc ;   m,  n,  and  o ; 

r,  s,  and  t. 
IS.  Add  If  m,  n,  o ;  add  u,  v,  w,  x. 

14.  Add  g,  h,  i,  J,  k,  I,  m,  n,  0 ;  add 

p,  q,  r,  s,  t,  u,  V,  u\  x,  ij,  z. 

15.  In  each  column  g,  h,  i,  etc.,  add 

all  the  numbers. 
How  many  are 

16.  U-g-ht 

17.  57-g-h,  etc.,  to  H 

18.  S5- g-h,  etc.,  tog'? 

19.  H-r-s-t,  etc.,  to2;? 

20.  lOO-g'-A-i,  etc.,  t0  2;? 


34 


MUL  TIP  Lie  A  TION. 


SECTION    lY. 

MULTIPLICATION. 
Multiplication  of  2's,  3's,  and  4's. 

95.  How  many  apples 
are  there  in  the  hand? 
in  the  basket?  on  the 
table  ?    How  many  apples 

in  ail  ? 

a.  How  many  are  three 
2's,  or  3  times  2  ? 

h.  How  many  are  four 
2's?   five  2's?   6  times  2? 

In  finding  the  answer  to  each  of  these  questions,  you 
united  two  or  more  equal  numbers. 

96.  Uniting  two  or  more  equal  numbers  is  multiplying, 

97.  One  of   the    equal  numbers   to   be   united   is   the 
xnultiplicsLiid. 

98.  The  number  that  tells  how  many  equal  numbers  are 
to  be  united  is  the  multiplier, 

99.  The  result  obtained  by  multiplication  is  the  product. 

c.  In  the  example  "  seven  2's  are  14,"  which  is  the  multi- 
plicand ?  the  multipher  ?  the  product  ? 

d.  What  is  the  product  of  2  multiplied  by  8,  or  8  times  2  ? 

100.  2  multiplied  by  8  may  be  written  2x8.      The 
sign  X  means  ''  multiplied  by." 

e.  What  is  the  product  of  2  x  9  ?   of  2  x  10  ? 

/.    Add  by  2's  from  2  to  24.    The  numbers  you  have  named 
are  all  the  products  of  2  up  to  24. 


(7) 

(8) 

(9) 

(iO) 

(H) 

2 

2 

2 

2 

2 

6 

11 

8 

12 

10 

MULTIPLICATION.  35 

Drill  Exercises. 

101.  g-  Write  the  products  found  by  multiplying  2's  from 
one  2  to  twelve  2's,  in  the  form  given  below.  table  * 

h.    Kepeat  from  memory  what  you  have       q^^    2    is    2 
written.  Two    2's  are  4. 

i.     Eepeat    the   table  several  times,  for-       Three  2's  are  6. 
ward  and  back.  Etc. 

Name  the  products  of  the  pairs  of  numbers  written  below 
till  you  can  give  them  in  any  order  at  sight. 

(!)        (2)       (5)       (A)       (5)       (6) 

j.   Multiply  2      2      2      2      2      2 
by        2      4      3      7      5      9 

k.  Multiply  these  numbers  again,  and  add  1  to  each  product. 

In  this  exercise,  think  what  the  products  are,  but  name  only  the 
final  results.  Thus,  in  multiplying  the  first,  think  4  and  then  say  5  ; 
in  multiplying  the  second,  think  8  and  then  say  9 ;  and  so  on. 

102.  L    Add  by  3's  from  3  to  36. 

m.  Write  the  table  of  3's  from  one  3  '^^^^^• 

to  twelve  3's.  ^"*^  2,    ^^   ^^ 

n.    Repeat  tlie  table  from  memory  for-  ^ 

ward  and  back. 

{1)       (2)        (3)        (A)      (5)        (6)       (7)        (5)        (9)        (10)      (11) 

o.  Multiply  33333333333 
by        2      5     10      3      8      4     11      6     12       7      9 

p.   Multiply  these  numbers  again  and  add  1  to  each  product. 
q.    Multiply  again  and  add  2  to  each  product. 

103.  r.    Add  by  4's  from  4  to  48. 

TABLE. 

s.   Write  the  table  of  4's  from  one  4  to         r^      a    •      * 

One  4    IS     4. 

twelve  4's.  Two  4's  are  8. 

t.    E-epeat  the  table  forward  and  back.  Etc. 

*  Sec  Appendix,  page  138. 


3  6  MUL  T I  PLICA  TION. 

(1)     (2)      (5)       (^)       (J)      (6)      (?)       (5)        (9)       (10)    (11) 

u.  Multiply  44444444444 
by        3      5      2       7      9      4    10     12      6       8     11 

V.    Multiply  these  numbers  again  and  add  1  to  each  product. 
w.  Multiply  and  add  2.  x.    Multiply  and  add  3. 

104.    Oral  Examples. 

a.  If  a  chair  has  2  arms,  how  many  arms  have  3  chairs  ? 

4  chairs  ? 

Solution.  —  If  a  chair  has  2  arms,  3  chairs  will  have  3  times  2 
arms,  which  is  6  arms.     Aiis.  6  arms.     (See  Appendix,  page  139.) 

b.  "If  a  knife  has  3  blades,  how  many  blades  have  2  knives? 
3  knives  ? 

c.  If  each  of  Harry's  6  playmates  gave  him  3  peaches,  how 
many  peaches  did  he  receive  ? 

d.  A  fork  has  4  tines.  How  many  tines  have  2  forks  ?  3 
forks  ?   7  forks  ? 

e.  At  $  4  a  week,  how  many  dollars  will  a  person  earn  in  4 
weeks  ?    in  5  weeks  ?    in  10  weeks  ? 

/.    What  cost  6  mats  at  $  1  each  ?  at  $  2  ?   at  $3  ?  at  $4  ? 

g.  One  dollar  equals  2  half  dollars.  How  many  half  dollars 
do  2  dollars  equal  ?    3  dollars  ?    5  dollars  ? 

iz.  A  square  has  4  sides.     How  many  sides  have  3  squares  ? 

i.  A  triangle  has  3  angles.  How  many  angles  have  2  tri- 
angles ?   3  triangles  ?    8  triangles  ? 

J.  If  a  pear  is  cut  into  3  equal  parts,  each  of  the  parts  is 
1  third.     How  many  thirds  in  4  pears  ?  in  5  pears  ? 

105.  k.  How  many  are  three  2's  ?  two  3's  ?  Is  there  any 
difference  in  the  answers  ? 

1.  How  many  are  2  X  4  ?  4x2?  four  3's  ?  three  4's? 
222.  How  many  are  3  x  2  ?  3  tens  x  2  ?  3  hundreds  x  2  ? 
22.  How  many  are  3x5?  3  tens  x  5  ?  3  hundreds  x  5  ? 
o.  How  many  are  4  x  7  ?  4  tens  x  7  ?  4  hundreds  x  7  ? 
p.    How  many  are  4x9?   4  tens  x  9  ?   4  hundreds  x  9  ? 


MUL  TIPLICA  TION.  37 

Examples  for  the  Slate. 

106.  Copy  the  following  examples  on  your  slate.    Begin- 
ning  with    the    units,   midti2oly    the 
Q.  -Qo        units,  the  tens,  and  the  hitndreds  scjp- 

2  3        arately,  writing  the  products  as  they 

—  are  written  in  Examples  a  and  b. 

(1.)  (2.)  (3.)  (4.)  (5.) 

Multiply    313  423  304  402  404 

by  3  3  4  5  6 

c.  d.  In  Example  c,  the  product  of 

Multiply       43  $2.40       the   units   is    18.       18   units    are 

"j^y  g  7       equal  to  1  ten  and  8  units.    We 

o^         ^  A~Qn       ^^'i^ite  8  under  the  line  and  keep  the 

1  ten  to  add  to  the  product  of  the 

tens.     The  entire  product  is  258.    In  Example  d,  as  the  multiplicand 

is  dollars  and  cents,  so  the  product  is  dollars  and  cents. 

For  the  sake  of  rapid  working,  use  as  few  words  as  possible.  Thus, 
in  Example  c,  say  ''^eighteen,  twenty-four,  twenty-five^  While  saying 
"  eighteen,^^  write  8 ;  and  while  saying  "  twenty-five,"  write  5  and  2. 


(6.) 

(7.) 

(8.) 

(9.) 

(10.) 

(11.) 

Multiply  $413 

$423 

$  1.42 

$3.14 

$3.24 

$2.30 

by              7 

10 

11 

9 

8 

12 

12.  Mr.  Granger  has  115  sheep  and  Mr.  Oaks  has  3  times 
as  many.     How  many  sheep  has  Mr.  Oaks  ? 

13.  After  travelhng  134  miles  on  a  journey,  Mr.  Niles  had 
5  times  that  distance  to  go.     How  many  miles  had  he  to  go  ? 

14.  A  man  bought  a  boat  for  $  132  and  a  horse  for  4  times 
that  sum.     What  did  he  pay  for  his  horse  ? 

15.  What  will  8  hats  cost  at  $  2.40  apiece  ? 

16.  What  will  9  caps  cost  at  $  1.34  apiece  ? 

17.  What  will  12  chairs  cost  at  %  4.42  apiece  ? 


38  MULTIPLICATION. 

Multiplication   of  5's  and  G's. 

Drill  Exercises. 

107.  a.    Add  by  5's  from  5  to  60. 

b.  Write  the  table  of  5's  from  one  5  to  twelve  5's  in  the 
same  way  that  you  wrote  the  tables  of  2's,  3'S;  and  4's. 

c.  Eepeat  the  table  forward  and  back. 

{!)       (^)       {3)       U)       (5)       {6)        (7)        (S)       {9)     (10)     {11) 

d.  Multiply     555555      5555      5 

by  3      8      7      2      9      4     10    12      6    11      5 

e.  Multiply  these  numbers  again  and  add  1  to  each  product. 
/.    Multiply  and  add  2.  g.    Multiply  and  add  3. 

h.    Multiply  and  add  4. 

108.  i.    Add  by  6's  from  6  to  72. 

J.    Write  the  table  of  6's  from  one  6  to  twelve  6's. 
k.  Eepeat  the  table  forward  and  back. 

{1)        (^)      (5) 

L    Multiply    6      6      6 
by         2      7    10 

222.  Multiply  these  numbers  again  and  add  1  to  each  product. 
22.    Multiply  and  add  2.  p.    Multiply  and  add  4. 

o.    Multiply  and  add  3.  q.    Multiply  and  add  5. 

109.     Oral  Examples. 

r.  If  there  are  5  books  on  the  table  and  10  times  as  many 
on  the  shelf,  how  many  are  there  on  the  shelf  ? 

s.  There  are  6  working  days  in  one  week.  How  many  are 
there  in  2  weeks  ?   in  5  weeks  ?   in  10  weeks  ? 

t.  A  cube  has  6  faces.  How  many  faces  have  3  cubes  ? 
6  cubes  ?    12  cubes  ? 

u.  If  five  men  can  do  a  piece  of  work  in  2  d^ys,  in  what 
time  can  1  man  do  it  ? 


(^) 

(5)       (6) 

(7) 

{8) 

(9) 

(10) 

ill) 

6 

6      6 

6 

6 

6 

6 

6 

3 

8    11 

4 

9 

12 

5 

6 

MULTIPLICATION,  39 


110. 

Examples 

for  the 

Slate. 

(18.) 

(19.) 

(20.) 

(2i.) 

(22.) 

Multiply 

456 

564 

645 

450 

536 

by 

2 

■       3 

4 

5 

6 

(23.) 

(24.) 

(25.) 

(26.) 

(27.) 

(28.) 

524 

650 

15.46 

16.34 

$4.65 

$5.42 

7 

8 

9 

10 

11 

12 

29.  There  are  365  days  in  a  common  year.  How  many 
days  are  there  in  3  common  years  ? 

30.  There  are  366  days  in  a  leap  year.  How  many  days 
are  there  in  5  common  years  and  1  leap  year  ? 

31.  At  $  6.Q>b  a  week  for  board,  what  is  the  cost  of  7  weeks' 
hoard  ? 

32.  At  $6.35  a  day,  what  are  Mr.  Dole's  wages  for  6 
days  ? 

33.  Charles  has  $  6.40  and  his  brother  has  8  times  as  much. 
How  much  money  has  his  brother  ? 

34.  Mary  bought  12  yards  of  silk  at  $  1.65  a  yard.  What 
did  it  cost  her  ? 

35.  What  must  be  paid  for  11  acres  of  land  at  $2.56  an 
acre  ? 

36.  By  working  9  hours  a  day,  a  man  can  do  a  certain  piece 
of  work  in  156  days.  How  many  days  will  it  take  him  if  he 
works  1  hour  a  day  ? 

37.  If  6  men  can  build  a  fence  in  46  days,  in  how  many 
days  can  1  man  build  it  ? 

38.  How  many  are  10  times  650  trees  ?    12  times  $  536  ? 

39.  Multiply  456  by  3  and  by  4  and  add  the  products. 

40.  Multiply  564  by  5  and  by  6  and  add  the  products. 

41.  Multiply  654  by  7  and  by  8  and  add  the  products. 

42.  Multiply  346  by  9  and  by  11  and  add  the  products. 

43.  Multiply  Q>QQ>  by  4  and  by  6  and  add  the  products. 


40  MUL  T I  FLIC  A  TION. 

Multiplication  of  7's  and  8's. 
Drill  Exercises. 

111.  a.    Add  by  7's  from.  7  to  84. 

b.  Write  tlie  table  of  7's  from  one  7  to  twelve  7's. 

c.  Eepeat  the  table  forward  and  back. 

(i)       (2)       (3)       {U)       (v)       (6)       (7)        {8)       (9)       {10)       (11) 

d.  Multiply  77777777777 

by         3       6      4      2      7      5    11      9    12     10       8 

e.  Multiply  these  numbers  again  and  add  2  to  each  product. 
/.    Multiply  and  add  3.  h.   Multiply  and  add  5. 

g.   Multiply  and  add  4.  i.    Multiply  and  add  6. 

112.  J.  Add  by  8's  from  8  to  96. 

k.  Write  the  table  of  8's  from  one  8  to  twelve  8's. 
1.    Eepeat  the  table  forward  and  back. 

(i)       (?)        (.5)       (A)        (5)        (6)        (7)        [8)        (9)       (10)     (11) 

222.  Multiply  8,8888       88      88       88 
by         7       9      2      8      3     10      4     11      5     12      6 

22.    Multiply  these  numbers  again  and  add  3  to  each  product. 
o.    Multiply  and  add  4.  q.    Multiply  and  add  6. 

p.    Multiply  and  add  5.  r.    Multiply  and  add  7. 

113.     Oral  Examples. 

s.    At  7  cents  a  yard,  what  will  4  yards  of  ribbon  cost  ? 

t.  There  are  7  days  in  a  week.  How  many  days  are  there 
in  2  weeks  ?    in  3  weeks  ?    in  5  weeks  ?    in  9  w^eeks  ? 

iz.  There  are  8  quarts  in  1  peck.  How  many  quarts  are 
there  in  2  pecks  ?   in  3  pecks  ?    in  4  pecks  ? 

V.  John  is  8  years  old,  and  his  father  is  five  times  as  old. 
How  old  is  his  father  ? 

w.  Annie  has  made  a  quilt  with  8  rows  of  squares  in  it,  8 
squares  in  each  row.    How  many  squares  are  there  in  the  quilt  ? 


MUL  TIPLICA  TION.  41 

X.  If  8  quires  of  paper  can  be  bouglit  for  $  1,  how  much  can 
be  bought  for  $6?   for  $7? 

y.  If  a  piece  of  work  can  be  done  in  2  days  by  working  8 
hours  a  day,  in  how  many  days  can  it  be  done  by  working  1 
hour  a  day  ? 

z.   What  is  the  product  ofTxlO?   7x11?   8x7?   7x8? 

114.  Examples  for  the   Slate. 

(44.)  (45.)           (46.)            (47.)  (48.) 

Multiply    678  875             748             837  728 

by    ^         2  3                  4                  5  6 


(49.) 

(50.) 

(61.) 

(52.) 

(53.) 

(54.) 

187 

780 

878 

787 

684 

768 

7 

8 

9 

10 

11 

12 

bb.   At  $  2  each,  what  will  187  pear-trees  cost  ? 

At  $2  each,  187  pear-trees  will  cost  187  times  $2.  But  187  times 
$2  is  the  same  as  2  times  $  187.  (Art.  105.)  Hence  we  multiply  187 
by  2.     Ans.  $374. 

6Q>.  At  1 3  a  day,  what  will  a  person  receive  for  278  days' 
work  ? 

57.  At  6  cents  a  pound,  what  will  168  pounds  of  sweet 
potatoes  cost  ? 

6^.  At  11  cents  a  pound,  what  will  287  pounds  of  sugar 
cost? 

59.  What  will  2  boxes  of  soap  cost,  each  box  contilining  74 
pounds,  at  9  cents  a  pound  ? 

60.  How  many  days  are  there  in  48  weeks  and  1  day  ? 

61.  At  8  cents  a  yard  for  muslin,  what  is  the  cost  of  4 
pieces,  47  yards  in  a  piece  ? 

62.  Multiply  278  by  6,  and  multiply  the  product  by  5. 

63.  Multiply  426  by  3,  and  multiply  the  product  by  7. 

64.  Multiply  177  by  4,  and  multiply  the  product  by  9. 

65.  Multiply  387  by  2,  and  multiply  the  product  by  11. 


42  '         J/  UL  T I  FLIC  A  TION. 

Multiplication   of  9's   and  lO's. 
Drill  Exercises. 

115.  a.  Add  by  9's  from  9  to  108. 

h.    Write  the  table  of  9's  from  one  9  to  twelve  9's. 

c.  Repeat  the  table  forward  and  back. 

(i)        (^)       (5)       U)       (J)       (6)      (7)       {S)      {[))      (10)    ill) 

d.  Multiply      999999999     9     9 

by         11      3      9      2      8    12     4      7    10     5      6 

e.  Multiply  these  numbers  and  add  4  to  each  product. 
/.  Multiply  and  add  5.  h.  Multiply  and  add  7. 
g.   Multiply  and  add  6.  i.    Multiply  and  add  8. 

116.  j.    Write  the  table  of  lO's  from  one  10  to  twelve  lO's. 
k.    Repeat  the  table  forward  and  back. 

(i)       (S)      (5)       iU)       (5)       (G)      {7)      {8)      [9)      {10)     [11) 

L    Multiply   10    10    10    10    10    10    10    10    10    10    10 
by  3    10    12      2    11      9      4      7      5      8      6 

222.  Multiply  these  numbers  and  add  5  to  each  product. 
22.    Multiply  and  add  6.  p.    Multiply  and  add  8. 

o.    Multiply  and  add  7.  q.    Multiply  and  add  9. 

117.  Write  upon  your  slate  the  numbers  from  1  to  9,  as 
they  are  written  below.     (See  Appendix,  p.  139.) 

Beginning  at  the  right, 

r.     Multiply  each  number  by  2  and  add  1  to  the  product. 
s.    Multiply  by  3  and  add  2.     w.  Multiply  by  7  and  add  6. 
t.    Multiply  by  4  and  add  3.     x.  Multiply  by  8  and  add  7. 
u.    Multiply  by  5  and  add  4.     y.   Multiply  by  9  and  add  8. 
V.    Multiply  by  6  and  add  5.     z.    Multiply  by  11  and  add  9. 


MUL  T I  PLICA  TION.  43 

118.     Oral  Examples. 

a.  If  a  man  works  9  hours  a  day,  how  many  hours  does 
he  work  in  2  days  ?   in  3  days  ?    in  4  days  ? 

b.  Mary  pays  10  cents  a  yard  for  braid.  How  much  does 
she  pay  for  4  yards  ?    for  5  yards  ? 

c.  If  you  sleep  9  hours  a  day,  how  many  hours  do  you  sleep 
in  6  days  ?    in  a  w^eek  ? 

d.  At  10  cents  apiece,  what  will  6  tops  cost  ?  What  will  7 
tops  cost?   8?   11?   12? 

e.  How  many  dimes  are  there  in  $  1  ?    in  $  6  ? 

/.  In  one  eagle  there  are  $  10.  How  many  dimes  are 
There  ? 

g.   What  is  the  product  of  9x8?   9x10?   9x12? 

h.   What  is  the  product  of  10  x  11  ?   10  x  12  ? 

i.  Is  there  any  difference  in  the  number  of  trees  in  two 
orchards,  if  one  has  9  rows  of  10  trees  each,  and  the  other 
10  rows  of  9  trees  each  ?     Why  ? 

j.    How  many  tenths  in  one  ?    in  two  ?    in  ten  ? 

119.    Examples  for  the  Slate. 

Q>Q.  Multiply  798  by  3  ;  by  4  ;  and  add  the  products. 
67.  Multiply  897  by  5  ;  by  6  ;  and  add  the  products. 
^^.    Multiply  590  by  7  ;  by  8;  and  add  the  products. 

69.  Multiply  489  by  9  ;  by  10  ;  and  add  the  products. 

70.  Multiply  397  by  11 ;  by  12  ;  and  add  the  products. 

71.  Multiply  1399  by  2  ;  by  3  ;  and  add  the  products. 

72.  Find  the  amount  which  must  be  paid  for  48  cords  of 
wood  at  $  6  a  cord,  and  96  cords  at  $  5  a  cord  ? 

73.  What  must  be  paid  for  2  barrels  of  sugar,  the  first  con- 
taining 249  pounds  at  8  cents  a  pound,  the  other  268  pounds 
at  11  cents  a  pound  ? 

74.  What  cost  498  tons  of  iron  at  $9  a  ton,  and  904  tons 
at  $  12  a  ton  ? 

For  other  examples  in  Multiplication  by  numbers  no  greater  than  12, 
see  page  71. 


44  MUL  T I  PLICA  TION. 

Multiplication  of  ll's  and  12's. 
Drill  Exercises. 

120.  a.   Add  by  ll's  from  11  to  132. 

b.  Write  the  table  of  ll's  from  one  11  to  twelve  ll's. 

c.  Eepeat  the  table  forward  and  back. 

(i)       {^)      (3)       (i)       (5)       (6)       (7)       {8)      (9)      {10)     (11) 

d.  Multiply  11    11    11    11    11     11    11    11    11    11    11 

by  7      4      8      3      9      2    10      5    11      6    12 

^  —  

e.  Multiply  these  numbers  and  add  5  to  each  product. 
/.  Multiply  and  add  6.  h.  Multiply  and  add  8. 
g.    Multiply  and  add  7.  i.    Multiply  and  add  9. 

121.  j.   Add  by  12's  from  12  to  144. 

k.  Write  the  table  of  12's  from  one  12  to  twelve  12's. 
1.    Repeat  the  table  forward  and  back. 

(i)       (S)       {3)        (i)       (.5)        {6)       (7).      {8)       (9)     {10)     {11) 

m.  Multiply  12    12    12    12    12    12    12    12    12    12    12 
by  7    10      2    11      3    12      4      9      5     8      6 

22.  Multiply  these  numbers  and  add  5  to  each  product. 
o.    Multiply  and  add  6.  q.    Multiply  and  add  8. 

p.    Multiply  and  add  7.  r.    Multiply  and  add  9. 

122.    Oral  Examples. 

s.  If  it  taltes  11  yards  to  make  a  dress,  how  many  yards 
will  it  take  to  make  2  dresses  ?   3  dresses  ? 

t.  There  are  4  cousins  each  11  years  old,  what  is  the  sum 
of  their  ages  ?  They  have  a  teacher  5  times  as  old  as  either 
of  them.     How  old  is  the  teacher  ? 

u.  In  one  foot  there  are  12  inches.  How  many  inches  are 
there  in  4  feet  ?    in  5  feet  ?    in  10  feet  ? 

V.  How  many  inches  are  there  in  a  yard,  which  is  3  feet  ? 
in  2  yards,  or  6  feet  ? 

w.  How  many  eggs  are  there  in  7  dozen  ?    in  10  dozen  ? 


a. 

b. 

c. 

Multiply    9 

9 

9 

by         10 

100 

1000 

MULTIPLICATION.  45 

123.    Examples  for  the  Slate. 

Ten  nines  equal  9  tens 
(Art.  105),  or  90.    So  100 
nines  equal  9  hundreds,  or 
~90        ~900  QQQQ         900 ;  and  1000  nines  equal 

9  thousands,  or  9000. 
Hence,  to  multiply  by  10,  annex  a  zero  to  the  multiplicand;  to  mul- 
tiply by  100,  annex  two  zeros;  to  multiply  by  1000,  annex  three  zeros. 

75.  Multiply  69  by  10  ;  69  by  100  ;  6  by  1000. 

76.  Multiply  47  by  10  ;  by  100  ;  and  add  the  products. 

77.  Multiply  8  by  100  ;  by  1000 ;  and  add  the  products. 

78.  Multiply  9  by  10 ;  by  100 ;  by  1000 ;  and  add  the  products. 

79.  Multiply  5  by  1000;   by  100;  by  10;    and  add   the 
products. 

80.  Multiply  42  by  10 ;    268  by  10 ;   3  by  1000 ;    and  add 

the  products. 

When  the  multiplicand  and  multi- 
124.    d.   Multiply  30       p^-g^,^  ^^  ^^^^^^  ^^  ^j^^j^^  ^^^^^  ^g^^g  ^^ 

^y        ^^       the  right  hand,  disregard  the  zeros  in 
1200     multiplying,  hut  annex  to  the  p^roduct  as 
many  zeros  as  were  disregarded. 

(81.)        (82.)        (83.)        (84.)         (85.)       (86.) 
Multiply  40  300  240  710  480  20 

by       60  2  20  30  30  400 

87.  When  hay  is  $  20  a  ton,  what  must  be  given  for  10 
tons  ?    for  100  tons?  for  20  tons  ? 

88.  There  are  20  things  in  a  score.     How  many  things  are 
there  in  100  score  ? 

89.  What  is  the  cost  of  20  rakes  at  35  cents  each  ? 

90.  Mr.  Oakes  sold  140  bushels  of  potatoes  at  60  cents  a 
bushel.     What  did  he  receive  for  them  ?  ^--«==ss=::=:= 

91.  How  many  men  working  1  hour  would  he^>^^^^^  ^o 
do  as  much  as  80  men  working  10  hours  ?         f^"^       ^^ 


46  MULTIPLICATION. 

92.  Multiply  23  by  2  and  by  30  and  add  the  products. 

93.  Multiply  42  by  4  and  by  50  and  add  the  products. 

94.  Multiply  bQ  by  3  and  by  20  and  add  the  products. 

95.  Multiply  61  by  5  and  by  50  and  add  the  products. 

96.  Multiply  72  by  6  and  by  50  and  add  the  products. 

97.  Multiply  37  by  4  and  by  70  and  add  the  products. 

98.  Multiply  36  by  3  and  by  40  and  add  the  products. 

99.  Multiply  95  by  5  and  by  80  and  add  the  products. 

100.  Multiply  48  by  7  and  by  60  and  add  the  products. 

101.  Multiply  29  by  4  and  by  30  and  add  the  products. 

126.    e.   Multiply  29  by  34 

WRITTEN   WORK. 

29  To  multiply  by  34,  multiply  first  by  4, 

34  then  by  30,  and  add  the  products. 

J        1      /(  ^^   multiplying   by   30,  the   product  is 

116  =  product  by  4.  g^^    ^^^^  .^  .^   ^^^  necessary  to  write  the 

87   -product  by  30. 


zero. 


986  =  product  by  34. 


(102.)   (103.)       If  you  have  made  no  mistake,  the  an- 
M  Ifi-nlv    S2          84     ^^^'srs  to  Examples  102  and  103  are  equal. 
,            QA          ^9      Hence,  to  prove  examples  in  multiplica- 
*^        tion,  multiply  the  multiplier  by  the  multi- 
plicand.   The  two  products  ought  to  he  equal. 

Multiply  and  prove  the  following  examples : 


(104.) 

(105.) 

(106.) 

(107.) 

(108.) 

(109.) 

34 

52 

76 

48' 

39 

83 

x45 

x37 

x58 

x64 

x25 

x98 

110.  How  many  are  67  x  79  ?  (114.)  94  x  79  -  what  ? 

111.  How  many  are  67  X  36  ?  (115.)  83  x  37  =  what  ? 

112.  How  many  are  52  x  45  ?  (116.)  79  x  67  -  what  ? 

113.  How  many  are  68  x  25  ?  (117.)  94  x  89  -  what  ? 


MULTIPLICATION.  47 


126.    /.   Multiply  145  by  27  ;  205  by  27. 

PROOF.  PROOF. 

27  27 

145  205 


135  =  product  by  5.  135  =  product  by  5. 

108    =  product  by  40.  54      -  product  by  200. 

2J_  =  product  by  100.  ^  ^  p^^^^^^  ^^  205. 
3915  =  product  by  145. 

118.  *  Multiply  184  by  47.       122.   How  many  are  583  x56? 

119.  Multiply  235  by  39.       123.   How  many  are  28  x  470  ? 

120.  Multiply  605  by  15.       124.   How  many  are  731  x  13  ? 

121.  Multiply  62  by  134.       125.   How  many  are  33  x  206  ? 

For  other  examples  in  Multiplication,  see  page  71. 

127.    Applications. 

126.  At  11  cents  a  pounds  what  will  a  barrel  of  269  pounds 
of  sugar  cost  ? 

127.  If  a  train  of  cars  runs  453  miles  in  a  day,  bow  many 
miles  will  it  run  in  21  days  ? 

128.  If  a  tailor  makes  432  coats  in  a  year,  and  puts  14  but- 
tons on  each  coat,  how  many  buttons  does  he  use  ? 

129.  Mr.  Frost's  cow  Lily  yielded  154  pounds  of  butter  in 
a  year.  The  butter  was  sold  at  35  cents  a  pound.  How  much 
money  did  Mr.  Frost  receive  for  the  butter  ? 

130.  A  party  of  23  persons  hired  a  schooner,  paying  $  3.75 
apiece  for  the  use  of  it.     How  much  did  all  pay  ? 

131.  What  must  be  paid  for  5  cheeses  weighing  85  pounds 
each,  at  16  cents  a  pound  ? 

132.  Twelve  dozen  make  a  gross.  Twelve  gross  make  a 
great  gross.  How  many  pens  are  there  in  a  great  gross 
of  pens  ? 

133.  A  family  of  4  boarded  in  Plymouth  for  9  weeks  at  the 
rate  of  $  6.50  a  week  for  each  person.  What  was  the  cost  of 
board  for  the  whole  time  ? 


48  MISCELLANEOUS  EXAMPLES. 

128.    Miscellaneous  Oral  Examples. 
Eepeat  the  following  tables : 

liiquid  Measure.  Dry  Measure. 

4  gills     =  1  pint.  2  pints    =  1  quart. 

2  pints    =1  quart.  8  quarts  =  1  peck. 

4  quarts  =  1  gallon.  4  pecks  =  1  bushel. 

a.  How  many  quarts  are  there  in  one  gallon  ?  how  many 
pints  ?    how  many  gills  ? 

b.  What  is  the  cost  of  a  gallon  of  milk  at  6  cents  a  quart  ? 

c.  If  a  gill  of  milk  is  required  for  a  pint  of  coffee,  how  many 
gills  are  required  for  a  quart  ?  for  a  gallon  ? 

d.  How  many  quarts  are  there  in  2  gallons  and  1  quart  ? 

e.  In  a  bushel  there  are  how  many  pecks  ?  how  many 
quarts  ?    how  many  pints  ? 

/.  At  10  cents  a  quart  for  tomatoes,  how  many  cents  will  a 
man  receive  for  1  bushel  and  2  pecks  ? 

g.  If  I  buy  3  bunches  of  asparagus  at  8  cents  a  bunch, 
2  quarts  of  new  potatoes  at  10  cents  a  quart,  and  some  lettuce 
for  13  cents,  how  many  cents  should  I  pay  for  them  ? 

h.  If  what  I  bought  came  to  57  cents,  and  I  gave  in  pay- 
ment a  1-dollar  bill,  how  much  change  should  I  receive  in 
return  ? 

Note.  To  take  57  cents  from  $  1,  or  100  cents,  take  away  jfirst  50  cents, 
and  then  take  7  cents  from  the  remainder. 

i.  John  was  sent  to  the  store  with  60  cents  to  buy  3  pounds 
of  beef  at  11  cents  a  pound,  6  pounds  of  rhubarb  at  2  cents  a 
pound,  and  2  bunches  of  radishes  at  5  cents  a  bunch.  How 
many  cents  should  he  bring  back  ? 

J.  Edith  bought  4  yards  of  cambric  at  12  cents  a  yard  and 
had  9  cents  left.     How  many  cents  had  she  at  first  ? 

k.  12  inches  make  a  foot  and  3  feet  make  a  yard.  How 
many  inches  make  a  yard  ? 


MISCELLANEOUS  EXAMPLES.  49 

L  Measure  your  height  in  inches,  also  measure  the  distance 
you  can  reach  with  your  arms  extended,  and  find  the  difference. 

Notp:.  In  doing  the  following  examples  write  the  numbers  on  your  slate, 
if  necessary. 

m.  A  traveller  bought  for  his  breakfast,  2  sandwiches  at  8 
cents  apiece,  a  cup  of  coffee  at  10  cents,  and  2  pears  at  6  cents 
apiece.     What  did  his  breakfast  cost  him  ? 

n.  For  his  dinner  he  bought  steak  for  30  cents,  2  bananas 
at  8  cents  apiece,  some  pie  for  10  cents,  and  a  cup  of  tea  for  5 
cents.     What  did  his  dinner  cost  him  ? 

o.    How  much  more  did  his  dinner  cost  than  his  breakfast? 

p.    How  much  did  his  breakfast  and  his  dinner  both  cost  ? 

q.  How  much  change  should  he  receive  if  he  gave  in  pay- 
ment for  his  breakfast  and  dinner  a  1-dollar  bill  ? 

T.  3  and  2,  multiplied  by  2,  less  6,  multiplied  by  3,  are  how 
many  ? 

s.   7  times  3,  plus  4,  plus  5,  less  6,  less  8,  are  how  many  ? 

t.   Take  14,  subtract  8,  add  3,  add  2,  multiply  by  2. 

u.  6,  and  4,  and  7,  and  5,  less  12,  multiplied  by  4  =  what  ? 

V.  Think  of  any  number  less  than  8,  multiply  that  number 
by  5,  subtract  3  times  the  number  thought  of,  add  2,  add  4, 
subtract  twice  the  number  thought  of.   What  number  have  you  ? 

129.     Examples  for  the  Slate. 

134.  In  a  quire  of  paper  there  are  24  sheets.  How  many 
sheets  are  there  in  20  quires  or  a  ream  ? 

135.  After  a  collection  was  taken  in  church,  one  box  con- 
tained $  25.34,  another  $  10.17,  another  $  14.21,  and  another 
$  15.98.     How  much  did  all  contain  ? 

136.  In  a  basket  of  eggs  I  counted  175.  S2  of  them  were 
turkey's  eggs  and  the  rest  were  hen's  eggs.  How  many  hen's 
eggs  were  there  ? 

137.  Charles  started  to  carry  a  basket  containing  132  eggs, 
but  he  fell  and  broke  17  turkey's  eggs  and  29  hen's  eggs. 
How  many  eggs  remained  whole  ? 


50  MISCELLANEOUS  EXAMPLES 

138.  Add  together  all  the  numbers  from  1  to  24. 

139.  There  were  landed  from  a  ship  38  boxes  of  lemons. 
If  each  box  contained  250  lemons,  how  many  lemons  were 
there  in  all  ? 

140.  In  an  orchard  of  125  trees,  the  owner  set  out  37  more, 
of  which  13  died.     How  many  remained  alive  ? 

141.  If  1  sparrow  destroys  235  caterpillars  in  a  day,  how 
many  will  6  sparrows  destroy  in  a  week  ? 

142.  How  many  panes  of  glass  are  there  in  a  house  which 
has  45  windows  of  6  panes  each,  and  4  cellar  windows  of  3 
panes  each  ? 

143.  A  hod-carrier  went  up  and  down  a  ladder  42  times  in  a 
day.  If  the  ladder  had  30  rounds,  how  many  steps  did  he  take 
upon  it  during  the  day  ? 

144.  The  children  of  a  school  can  be  arranged  in  24  rows, 
28  in  a  row.  How  many  more  children  would  be  needed  to 
make  27  rows,  27  in  a  row  ? 

145.  A  boy  bought  a  writing-desk  for  $2.75  and  a  jack- 
knife  for  $  1.37.  He  spoiled  his  jack-knife  in  mending  his 
writing-desk  and  then  sold  the  writing-desk  for  $  3.86.  Did 
he  gain  or  lose  by  the  whole  transaction,  and  how  much  ? 

146.  A  barrel  of  flour  contains  196  pounds.  If  a  pound  of 
flour  is  put  into  each  loaf  of  bread,  how  many  loaves  may  be 
made  from  25  barrels  of  flour  ? 

147.  If  7  persons  consume  a  barrel  of  flour  in  13  weeks,  how 
many  persons  will  consume  20  barrels  in  1  week  ? 

148.  A  man  bought  4  bushels  of  potatoes  for  $  3.50,  and  sold 
them  at  80  cents  a  peck.  How  much  did  he  receive  for  what 
he  sold  ?     How  much  did  he  gain  ? 

149.  A  grocer  paid  $  5.00  for  a  box  containing  294  oranges ; 
17  of  the  oranges  were  spoilt,  and  he  sold  the  remainder  at 
5  cents  apiece.     Did  he  gain  or  lose,  and  how  much  ? 

150.  A  dealer  counted  4  boxes  of  oranges,  and  found  in  the 
first  box  32  dozen  and  4 :  in  the  second,  35  dozen  and  3  ;  in 
the  third,  28  dozen  and  11 ;  and  in  the  fourth,  36  dozen  and 
7.     How  many  oranges  were  there  in  all  ? 


DIVISION. 


51 


SEOTIOIf   T. 


DIVISION. 
Division  by  2,  3,  and  4. 

130.  How  many  apples 
are  there  on  the  table  ? 

If  you  give  these  6  ap- 
ples to  some  children,  giv- 
ing them  2  apiece,  to  how 
many  children  will  you 
give  them  ? 

a.  How  many  2's,  or  how 
many  times  2,  are  there  in  6  ? 

131.  Finding  how  many  times  one  number  is  contained 
in  another  is  dividing. 

132.  The  number  to  be  divided  is  the  dividend. 

133.  The  number  by  which  we  divide  is  the  divisor. 

134.  The  result  obtained  by  dividing  is  the  quotient. 

b.  Divide  10  by  2,  that  is,  find  how  many  2's  there  are 
in  10. 

c.  What  is  the  quotient  of  18  divided  by  2  ?  of  20  divided 
by2? 

d.  If  you  have  11  apples,  to  how  many  children  can  you 
give  2  apiece,  and  how  many  apples  will  be  left  ? 

135.  The  part  of  the  dividend  left  after  the  equal  num- 
bers are  taken  away  is  the  remainder. 

e.  How  many  2's  are  there  in  13,  and  what  is  the  remainder  ? 


52  DIVISION, 

f.  What  is  the  quotient  of  2S  divided  by  4  ? 

136.    28  divided  by  4  may  be  written  thus,  28  ~  4.    The 
sign  -^  means  ''  divided  by." 

^.  What  is  the  quotient  of  36  -  4  ?  of  36  -  3  ? 
h.  What  is  the  quotient  of  27  --  3  ?  of  22  -  2  ? 
i.    What  is  the  quotient  of  13  ^  3,  and  what  remains  ? 

137.     Drill  Exercises. 

j\    Repeat  the  multiplication  table  of  2's. 

k.   How  many  2's  in  2  ?   4  ?   6  ?   8  ?   10  ?   12  ?   14  ?   16  ? 
18?  20? 

i.    Write  the  division  table  in  the  form  table.* 

given  in  the  margin^  from  2  in  2  to  2's  in 
24. 

222.  Eepeat  the  division  table  forward  and 
back. 

Divide  by  2  the  numbers  written  below,  naming  quotients 
and  remainders  at  sight. 

(i)      {2)        {3)         iU)         {5)        {6)        (7)        (8)        (9)       {10)       {11)      {12) 

n.  2)2     10     12      4      8       6     16     14     18     24     22     20 

o.   2)3    11     13      5      9      7    17    15    19    25     23     21 

138.    p.    Repeat  the  multiplication  table  of  3's. 

q.  Write  the  division  table  of  3's  from  3  in  3,  to  3's  in  36. 

r.    Repeat  the  table  forward  and  back. 

Divide  by  3  the  numbers  written  below : 

{!)         {3)       {3)        (^)       (.3)  (6)         (7)  {S)         {0)         {10)        {11)      {12) 

s.   3)  3  15  9  27  6  18  12  24  36  21  33  30 
t.  3)  4  17  11  2S     8  19  14  25  37  23  34  31 


2     in  2 

1. 

2's  in  4 

2. 

2's  in  6 

3. 

Etc. 

See  Appendix,  page  138. 


DIVISION.  ^^ 

139.    K.    Eepeat  the  multiplication  table  of  4's. 

V,    Write  the  division  table  of  4's  from  4  in  4  to  4's  in  48. 

w.  Eepeat  the  table  forward  and  back. 

Divide  by  4  the  numbers  written  below  : 

(i)        (2)      {3)        {U)       (5)         (6)        (7)        (S)         (9)       {10)      {11)       {IS) 

X.    4)16     8     4     24     12    28    20    32     40    48    36    44 
y,    4)  18   11     7     25     15    30    23    35    43    49     37    45 

140.    Oral  Examples. 

a.  How  many  tops  at  2  cents  each,  can  be  bought  for  6 
cents  ?   for  10  cents  ? 

h.  At  2  cents  each,  how  many  tops  can  be  bought  for  21 
cents,  and  how  many  cents  will  be  left  ? 

Sc/lution.^ —  As  many  tops  can  be  bought  for  21  cents  as  there  are 
2's  in  21;  which  is  10,  and  1  remains.     Ans.  10  tops;  1  cent  remains. 

c.  How  many  rows  of  chairs,  2  in  a  row,  can  you  make 
with  4  chairs  ?    12  chairs  ?    19  chairs  ? 

d.  How  many  quarts  are  there  in  10  pints  ?  in  14  pints  ? 
in  24  pints  ? 

e.  In  one  yard  there  are  3  feet.  How  many  yards  are 
there  in  6  feet  ?   in  9  feet  ?  in  10  feet  ? 

/.  At  3  cents  an  hour,  how  many  hours  must  you  work  to 
earn  15  cents  ?   24  cents  ?   30  cents  ? 

g.  Four  quarters  make  1  dollar.  How  many  dollars  in  8 
quarters  ?  in  12  quarters  ?  in  15  quarters  ? 

h.  How  many  gallons  are  there  in  12  quarts  ?  in  14  quarts  ? 

i.  How  many  bushels  are  there  in  4  pecks  ?  in  16  pecks  ? 
in  21  pecks  ? 

J.    How  many  pints  in  4  gills  ?    in  20  gills  ?    in  30  gills  ? 

k.  If  a  coat  can  be  made  from  4  yards  of  cloth,  how  many 
coats  can  be  made  from  24  yards  ?   from  29  yards  ? 

*  See  Appendix,  page  139. 


54 


DIVISION. 


Dividing  Numbers  into  Equal  Parts. 

141.  If  8  pears  be  divided 
equally  between  2  boys,  bow 
many  pears  will  each  boy  have? 
Each  boy  wdll  have  one  of 
the  two  equal  parts  into  which 
8  is  divided.  We  have  seen  by 
multiplication  that  2  fours  are 
8,  so  we  know  that  each  boy 
will  have  4  pears. 
142.    If  1  pear  be  divided  equally  between  two  boys, 

_     %^      what  will  each  boy  have  ? 

^^m^m^i^'  When  any  number  or  any  thing  is  di- 

'Llft  ■I^^Ip     vided  into  two  equal  parts,  the  parts  are 

called  halves.    When  it  is  divided  into  three  equal  parts, 

the  parts  are  called  thirds.     When  it  is  divided  into  four 

equal  parts,  the  parts  are  called  fourths;  and  so  on. 

143.     Oral  Exercises. 

a.  What  is  1  half  of  8  apples  ?  of  $  10  ?  of  14  ?  of  18  ? 

b.  What  is  1  third  of  6  pears  ?  of  $  9  ?  of  15  ?    of  27? 

c.  What  is  1  fourth  of  8  cents  ?  of  $  16  ?  of  12  ?  of  20  ? 

d.  If  12  figs  be  divided  equally  between  two  boys^  how 
many  figs  will  each  boy  receive  ? 

Solution.  —  Each  boy  will  receive  1  half  of  12  figs,  which  is  6  figs. 
Ans.  6  figs. 

e.  How  many  figs  w^ould  each  boy  receive  if  12  figs  were 
divided  equally  among  3  boys  ?   among  4  boys  ? 

/.  How  many  fishes  would  each  person  receive  if  24  fishes 
were  divided  equally  among  3  jDersons  ?    among  4  persons  ? 

g.  By  what  do  you  divide  to  find  1  half  of  a  number  ?  1 
third  ?    1  fourth  ? 

h.  How  many  are  9  -  3  ?   9  tens  -  3  ?   9  hundreds  -  3  ? 

i.    How  many  are  8^4?   8  tens  -^  4  ?   8  hundreds  ^  4  ? 

J.    How  many  are  27  -  3  ?   27  tens  -  3  ?   27  hundreds  -  3  ? 


DIVISION.  55 

Examples   for   the    Slate. 

144.   Copy  the  following  examples  on  your  slate.    Begin- 

iiinor  with  the   hundreds,  divide   the 
a.  ^  * 

Divisor  ^)  306  Dividend,     huudreds,  the  tens,  and  the  units  sep- 
~77j^  n   r   «      arately,  writing  the  quotient  as  it  is 
written  in  Example  a. 

Ill  doing  Example  a,  say  "  3's  in  3,  1;  in  0,  none;  in  6,  2." 
To  prove  your  work,  find  the  product  of  the  divisor  and  quotient. 
This  product  ought  to  equal  the  dividend. 

(1.)  (2.)  (3.)  (4.)  (5.) 

2)  842  2)  460  3)  366  3)  906  4)  480 

b.  c.  In  Example  b,  as  3  is  larger  than  1,  begin 

3)  1230  2)  1418       by  dividing  12  hundreds.     In  Example  c, 
~410  /TAQ       begin  by  dividing  14  hundreds  ;  and  when 

you  come  to  the  tens,  as  2  is  larger  than  1, 
put  a  zero  in  the  tens'  place  Crt'  the  quotient  and  divide  18  units. 

(6.)  (7.)  (8.)     .  (9.)  (10.) 

3)  1569    3)  2118    2)  1612    4)  1620    4)  8360 

"•  In  Example  d,  after  dividing  26  by  4,  there  is  a 

4)  426  — J  hem.      remainder  of  2,  which  is  written  at  the  right  of  the 
1Q5  dividend. 

Note.     In  proving  the  work,  the  remainder  must  he  added  to  the  product 
of  the  divisor  and  quotient. 

(11.)  (12.)  (13.)  (14.)  (15.) 

3)  2722         4)  2814         4)  3225         3)  2405         3)  3277 

16.  If  364  children  are  marching,  3  in  a  row,  how  many 
rows  are  there  ? 

17.  How  many  rows  would  there  be,  if  364  were  marching, 
4  in  a  row  ? 

18.  Albert  bought  4  canary  birds  for  $8.32.    What  was  the 
price  of  each  ? 

19.  A  father  left  $  8420  to  his  four  children.     What  was 
the  share  of  each  child  ? 


56  DIVISION. 

Division  by   5,  6,  and  7. 

145.    Drill  Exercises. 

a.  Repeat  the  multiplication  table  of  5's. 

b.  Write  the  division  table  of  5's  from  5  in  5  to  5's  in  60. 

c.  Repeat  the  table  forward  and  back. 

Divide  by  5  the  numbers  written  below  : 

{!)        (?)        {3)        (A)        (5)        {G)        (7)         (5)         (9)       {10)      {11)      {IS) 

d.  5)  10  30  50  25   5  35  m    15  40  60  20  45 

e.  5)  13  32  54  26   7  39  57  16  43  62  24  48 

146.  /.    Repeat  the  multiplication  table  of  6's. 

g.   Write  the  division  table  of  6's  from  6  in  6  to  6's  in  72. 
h.   Repeat  the  table  forward  and  back. 
Divide  by  6  the  numbers  written  below : 

{1)        {2)        {3)        {h)        {5)        {G)        (7)         {8)        {9)       {10)      {11)       {12 

L    6)  12  36  60  72   6  42  66     18  48  24  54  30 
j.   6)  15  '  37  64  77  11  44  68  23  49  29  58  34 

147.  k.    Repeat  the  multiplication  table  of  7's. 

1.    Write  the  division  table  of  7's  from  7  in  7  to  7's  in  84. 
m.  Repeat  the  table  forward  and  back. 
Divide  by  7  the  numbers  written  below  : 

{1)       p)  {3)       (/.)         (5)        {G)        (7)         {8)        (.9)       {10)      {11)     {12) 

n,   7)  28     49      7     63     77    14    m    70     21     84    35    42 
o.   7)  31     54    11     69     82     17     62     72     23    %S    36    46 

148.    Oral   Examples. 

p.  If  a  sack  can  be  made  out  of  5  yards  of  cloth,  how  many 
sacks  can  be  made  out  of  25  yards  ?    30  yards  ?    34  yards  ? 

q.  6  feet  make  a  fathom.  How  many  fathoms  deep  is  it  to 
a  rock  that  is  under  water  24  feet  ?    36  feet  ?    48  feet  ? 


DIVISION.  57 

T.  If  an  anchor  sinks  59  feet,  how  many  fathoms  does  it  sink 
and  how  many  feet  besides  ? 

s.  How  many  weeks  and  how  many  days  over  are  there  in 
25  days  ?    in  37  days  ?   in  60  days  ?   in  72  days  ? 

t.  What  is  1  fifth  of  2^  days  ?  1  sixth  of  $  36  ?  1  seventh 
of  84  cents  ? 

12.  Charles  bought  a  velocipede  for  $  15,  and  paid  1  fifth  of 
the  price  every  month.  Hai^tjnany  dollars  did  he  pay  a  month  ? 
How  many  months  did  it  take  him  to  pay  the  whole  ? 


149.    Examples  for  the  Slate. 

(20.) 

^      (21.)                    (22.) 

(23.) 

5)  1520 

6)  3541               6)  1809 

7)  6329 

^'  Ijli^ividing  17  (tens)  by  5,  we  have  3  (tens)  for  the 

5)  ilo-6        quotient,  and  2  tens  remain.     We  write  3  under  the 

34  line  in  the  tens'  place,  and  unite  the  2  tens  with  the  3 

units  of  the  dividend,  making  23  units. 

5's  in  23,  4  and  3  remain,  which  we  express  as  in  the  written  work 

above.     Ans.  34  and  3  remain. 

(24.)  {25.)  (26.)  (27.)  (28.)  (29.) 

5)184       ^5)283        6)728        7)1230        6)326        7)8142 

30.  How  many  sheep  at  $  6  apiece  can  be  bought  for  $  159, 
and  what  will  remain  ? 

31.  In  365  daj^s  how  many  weeks,  and  what  remains  ? 

32.  A  milkman  received  $86.45  for  milk  which  he  sold  at 
7  cents  a  quart.     How  many  quarts  did  he  sell  ? 

33.  How  man}^   settees  will  be    required  to  accommodate 
1000  persons  if  5  are  seated  on  each  settee  ? 

34.  Five  men  hired  a  boat  for  $  12.75,  agreeing  to  divide  the 
cost  equally.     What  was  the  share  of  each  ? 

35.  Six  men  shared  equally  a  profit  of  $  7464.     What  did 
each  receive  ? 

36.  What  is  1  fifth  of  1375  oxen  ?  1  seventh  of  $  4529  ? 


58  DIVISION. 

Division  by   8,   9,  and  10. 
Drill  Exercises. 

150.    a.    Repeat  the  multiplication  table  of  8's. 

b.  Write  the  division  table  of  S's  from  8  in  8  to  8's  in  96. 

c.  Kepeat  the  table  forward  and  back. 
Divide  by  8  the  numbers  written  below : 

(I)        (f)        (3)        (A)       (5)        {&)^fk        («)        (9)        {10)      (11)     (11) 


d.   8)  16  40  64  80   8  48  88  24  56    96  32 

72 

e.  8)  22  45  66    87  13  54  89  31  .^58  99  36 

75 

151.    /.    Eepeat  the  multiplication  table  of^'s. 

g.  Write  the  division  table  of  9's  from  9  in  9  to  9's  in  108. 

h.    E-epeat  the  table  forward  and  back. 

Divide  by  9  the  numbers  written  below  i^*^ 

(1) 


(2) 

(3) 

(^) 

(5) 

(G) 

(7) 

(8) 

(9) 

(10) 

(11) 

90 

36 

72 

54 

99 

18. 

63 

81 

27 

108 

i.   9)  45 

J.   9)  51  92  42  76  61  102  21  65  "^P  32  112 

152.  k.  Repeat  the  multiplication  table  oi  10^ 

1.   Write  the  division  table  of  K|'s  from  10  in  10  to  iO's  in  120. 
222.  Repeat  the  table  forward  and  back. 
Divide  by  10  the  numbers  written  below : 

(1)       (2)        (3)       (A)        (5)       (6)        (7f      (8)         (9)         (10)        (11) 

22.  10)  20    31    45    60    57     73     88     90     108    123    110 

Oral  Examples. 

153.  o.   Divide  9  yards  of  ribbon  equally  among  8  girls. 
Divide  10  yards. 

If  9  yards  are  divided  among  8  girls,  each  girl  will  receive  1  yard, 
and  there  will  still  be  1  yard  to  be  divided. 

If  1  yard  is  divided  amoncr  8  girls,  each  girl 

I      I       I       I       I       I      I      I       I  -^  r>        n  7  n 

will  receive  1  eighth  of  a  yard.     So  each  girl 
1  eighth.  'vvill  have  1  and  1  eighth  yards. 


DIVISION,  5^^ 

If  10  yards  are  divided  among  8  girls,  after  giving  1  yard  to  each 
.■..■■,,.     girl,  there  will  be  2  yards  to  be  divided,  which 
1  eigjjth.  will  give  2  eighths  of  a  yard  apiece.    So  each 

I    I    1    I    I    I    I    I    I     girl  will  have  1  and  2  eighths  yards. 

'  ''^'^'^-  p.   8  men  bought  a  fishing-net  for  $  21. 

What  part  of  the  price  should  each  pay  ?    How  many  dollars  ? 
q.  What  is  1  eighth  of  20  ?   of  23  ?   of  34  ? 
T.    Divide  41  rods  of  land  into  8  equal  parts ;    divide  52 
rods  ;  divide  78  rods. 

5.    If  a  party  of  10  men  catch  ^Q  pounds  of  fish,  what  is  the 
share  of  each  ? 

154.    Examples  for  the  Slate. 

a.    Divide  $  6530  by  8.  in  the  second  form  of  the  written 

wPviTTEN  WORK.  work  of  this  example,  the  last  re- 

^  8)  $  6530  -  $  2.          8)  $  6530  mainder,  $  2,  is  divided,  giving  a 

~r^7^                            ^Qifa  quotient  of  2  eighths,  which  is 

^  written  thus,  |.     Ans.  $816|. 

Divide  in  the  same  way  the  following  examples : 

(37.)  (38.)  (39.)  (40.) 

8)  3297  8)  4542  8)  4125  8)  6321 

(41.)  (42.)  (43.)  (44.) 

9)  7234  9)  1928  9)  1234  9)  7361 

(45.)  (46.)  (47.)  (48.) 

8)  5187  9)  1904  -10)  8761  10)  3205 

49.  Mr.  Eustis  bought  a  house  for  $  3824  and  paid  1  eighth 
of  it  every  year  till  it  was  paid  for.   How  much  did  he  pay  a  year  ? 

50.  Mr.  Clark  teaches  9  months  in  the  year  at  a  salary  of 
$  1900.     What  is  that  a  month  ? 

51.  Miss  Alvord  teaches  10  months  in  the  year  at  a  salary 
of  $  1275.     What  is  that  a  month  ? 

52.  If  it  costs  a  family  of  8  persons  $  1535  a  year  to  live, 
what  is  the  average  cost  for  1  person  ? 


60  DIVISION. 

Division  by  11  and  12. 

Drill  Exercises. 
156.    a.    Repeat  the  multiplication  table  of  ll's. 

b.  Write  the  division  table  of  ll's  from  11  in  11  to  ll's 
in  132. 

c.  Eepeat  the  table  forward  and  back. 
Divide  by  11  the  numbers  written  below : 

{1)        (^)        (5)        (>i)         (5)         (6)        (7)  [8)         (9)       {10)        [11) 

d.  11)  22  33  66    88  110  44  77  121  55  99  132 

e.  11)  25  37  71  95  118  53  81  122  65  101  141 

166.    /.    Eepeat  the  multiplication  table  of  12's. 
g.    Write  the  division  table  of  12's  from  12  in  12  to  12's 
in  144. 

h.    Eepeat  the  table  forward  and  back. 
Divide  by  12  the  numbers  written  below  : 


I   12)36 

iS)        {3) 

24  96 

144 

(•5)    (6) 

132  48 

(7)    [8) 

120  60 

(9)  {10)      {11) 

108  84  72 

J.   12)  42 

31  101 

152 

141  52 

122  71 

113  93  73 

167.    Oral  Examples. 

k.  At  12  cents  apiece,  how  many  melons  can  be  bought  for 
24  cents  ?   for  40  cents  ?   for  50  cents  ? 

i.  It  takes  Mary  60  hours  to  crochet  a  jacket.  How  many 
days  will  it  take  her  if  she  works  12  hours  a  day  ? 

221.  How  many  days  will  it  take  Mary  to  crochet  the  jacket 
if  she  works  11  hours  a  day  ? 

21.  How  many  feet  are  there  in  12  inches  ?  in  24  inches  ? 
in  18  inches  ?    in  96  inches  ?     (See  Example  u,  page  44.) 

o.  When  10  yards  of  silk  can  be  bought  for  $  15,  what  is 
the  price  of  1  yard  ? 

p.  There  are  12  months  in  a  year.  In  36  months  how 
many  years  ?    in  30  months  ?   in  42  months  ? 


DIVISION.  61 

q.  What  is  1  eleventh  of  $  22  ?   of  $  44  ?   of  $  66  ? 

r.  What  is  1  eleventh  of  $  27  ?   of  $  51  ?   of  $  72  ? 

s.  What  is  1  twelfth  of  36  n   of  60  /  ?   of  75  /  ? 

t.  What  is  1  twelfth  of  35  days  ?   of  100  days  ? 

158.    Examples  for  the  Slate. 

Find  how  many  times  $  11  is  contained  in  the  sums  of  money 
written  below  : 

(530  (54.)  {66.)  (56.) 

$  11)  $  5602     1 11)  $  6281     $  11)  $  3406     $  11)  $  4987 

Find  1  twelfth  of  the  sums  of  money  written  below : 
(57.)  (58.)  (59.)  (60.) 

12)  $  7681  12)  $  3908  12)  $  85.29  12)  $  50.03 

For  other  examples  in  Division  by  numbers  no  greater  than  12,  see 
page  71. 

159.    a.   Divide  1698  by  32. 
WRITTEN  WORK.  In  this  example,  since  32  is  a  larger  number 

32)  1698  (53 A       *^^^^  ^^^  ^^  ^^'^^  divide  169  (tens)  by  32. 

-j^^Q  There  are  about  as  many  32's  in  169  as  there 

are  3's  in  16,  which  is  5.     We  write  5  for  the 

tens  of  the  quotient,  at  the  right  of  the  curved 

^  line.     Multiplying  32  by  5  (tens),  we  have  160 

2  (tens).     Subtracting  160  from  169,  we  find  9  tens 

remain.     Uniting  with  these  9  tens  the  8  units 

of  the  dividend,  we  have  98  units. 

32's  in  98,  3  and  2  units  remain.  We  write  3  for  the  units  of  the 
quotient.  Dividing  the  remaining  2  by  32,  we  have  2  thirty-seconds, 
and  the  entire  quotient  is  53^^. 

Divide  in  the  same  way  the  following  examples : 
(61.)  (62.)  (63.)  (64.) 

_  21)  1164  (  42)  1984  (  44)  3668  (  51)  4897  ( 

k(65.)  (66.)  (67.)  (68.) 

3)  2998  (  62)  4528  (  67)  2363  (  41)  3784  ( 


62  DIVISION. 

69.  How  many  are  394  -  34  ?  73.  Divide  3949  by  84. 

70.  How  many  are  538  -  47  ?  74.  Divide  7009  by  51. 

71.  How  many  are  784  -  26  ?  75.  Divide  7958  by  68. 

72.  How  many  are  892  -  75  ?  76.  Divide  9876  by  92. 

160.    b.   Divide  1792  by  24. 

WRITTEN  WORK.  j^  Example  b,  when  we  say  that  there  are  about 

24)  1792  (74if  as  many  24's  in  179  as  2's  in  17,  we  find  8  for  the 

168  first  term  of  the  quotient;  but  on  multiplying  we 

~Tio  get  192,  a  number  too  large  to  subtract  from  179. 

Therefore  8  is  too  large  for  the  first  term  of  the 

quotient,  and  we  try  7,   as   shown  in   the   work 

16  above. 


77.  How  many  are  1443  -  74  ?  81.  Divide  9278  by  87. 

78.  How  many  are  2938  -  77  ?  S2.  Divide  2876  by  43. 

79.  How  many  are  8642-46  ?  83.  Divide  6444  by  96. 

80.  How  many  are  4672  -  58  ?  84.  Divide  5105  by  64. 

To  divide  by  10,  100,  or  1000. 

161.    c.    In  1456;,  how  many  tens  and  what  remains  ? 

Ans.  145  tens  and  6  remain. 

d.  In  1456j  how  many  hundreds  and  what  remains  ?   how 
many  thousands  and  Avhat  remains  ? 

e.  Divide  1456  by  10 ;    by  100  ;   by  1000. 

To  divide  by  10,  put  a  point  in  tlie 
1456 -f- 10  =145.6  dividend  one  place  to  the  left;  to  divide  by 
1456-^100  =14.56  100,  put  a  point  two  places  to  the  left;  to 
1456  -=- 1000  =  1.456  divide  by  1000,  put  a  point  three  ptlaces  to 
the  left. 
In  Example  e,  the  first  answer  is  145  and  6  tenths ;  the  second  is 
14  and  56  hundredths ;  and  the  third  is  1  and  456  thousandths. 

85.  Divide  3456  by  100.      88.  Divide  $  8100  by  10 ;  by  100. 

86.  Divide  252  by  100.       89.  Divide  $1649  by  100;  by  10. 

87.  Divide  $  3000  by  10.     90.  Divide  $  4930  by  1000. 


DIVISION.  63 

To    divide  United    States    Money,   carrying  the  Division    to 

Cents. 

162.    /.     Divide  $  34.87  by  32.     g.   Divide  $45  by  32. 


WRITTEN 

WORK. 

In  Example   /, 

/. 

s- 

the  dividend  has  2 

32)  $  34.87 

($ 

1.08§i 

32)  i 

>45  ($ 

1.40§J 

places  for  cents,  so 

32 

32 

in  the  quotient  we 

287 

1300 

mark  off  2  places 

256 

128 

for  cents. 

31 

20 

In  Example  g. 

after    the    dollars 

are  divided  there  is  a  remainder  of  $  13,  which  equals  1300  cents. 
1300  cents  divided  by  32  gives  40|^  cents,  and  the  whole  quotient 
is$1.40ff 

In  doing  the  following  examples,  whenever  a  remainder  is  dollarSj 
annex  two  zeros  and  continue  the  division  to  cents. 

91.  Divide  $  48.40  by  15.  97.  Divide  $  48  by  14. 

92.  Divide  $  82.36  by  32.  98.  Divide  $  28  by  32. 

93.  Divide  $  97.48  by  63.  99.  Divide  $  42  by  34. 

94.  Divide  $  99.19  by  42.  100.  Divide  $  84  by  41. 

95.  Divide  $  49.32  by  54.  101.  Divide  $  4  by  25. 

96.  Divide  $  85.12  by  81.  102.  Divide  $  8  by  27. 

163.   Applications. 

103.  How  many  piles  of  20  half-dollars  each,  can  you  make 
with  348  half-dollars  ? 

104.  How  many  piles  of  40  quarter-dollars  each,  can  you 
make  wdth  583  quarters  ? 

105.  Mr.  Goss  planted  1296  hills  of  corn  in  rows  of  27  hills 
each.     How  many  row^s  did  he  plant  ? 

106.  If  Mr.  Goss  had  planted  the  same  number  of  hills  in 
16  rows,  how  many  hills  would  he  have  had  in  a  row  ? 

107.  Mr.  Green  earned  $  28.68  in  a  month  and  spent  1 
twelfth  of  what  he  earned  for  books.  How  much  did  he  spend 
for  books  ? 


64  DIVISION. 

108.  A  man  bought  a  house  for  $2175,  agreeing  to  pay 

1  tenth  of  the  price  every  year  till  it  was  paid  for.  How  much 
did  he  agree  to  pay  each  year.  How  many  years  would  it  take 
him  to  pay  the  debt  ? 

109.  How  many  barrels  can  be  filled  with  2000  pounds  of 
flour,  each  barrel  containing  196  pounds  ? 

110.  A  load  of  potatoes  weighing  742  pounds  was  put  into 
bags  of  2  bushels  each.  If  each  bushel  weighed  60  pounds, 
how  many  full  bags  were  there,  and  how  many  pounds  re- 
mained ? 

111.  How  many  dollars  are  there  in  2762  cents  ?  in  5648 
cents  ? 

112.  If  a  man  walks  25  miles  a  day,  how  many  days  will  it 
take  him  to  walk  1000  miles  ? 

113.  There  are  5280  feet  in  a  mile.  How  many  times  must  a 
wagon-wheel  turn  in  going  a  mile  if  it  is  14  feet  round  the  rim  ? 

For  other  examples  in  Division,  see  page  71. 

Miscellaneous   Oral  Examples. 

164.  a.  Ira  sells  cherries  at  10  cents  a  quart.  How  many 
quarts  must  he  sell  to  receive  50  cents  ?    90  cents  ?    $  1  ? 

b.  Carl  sells  berries  at  9  cents  a  quart.  How  many  quarts 
must  he  sell  to  get  enough  money  to  buy  a  hat  worth  70  cents 
and  a  slate  worth  11  cents  ? 

c.  Jane  makes  paper-bags  at  8  cents  an  hour.  How  much 
will  she  earn  in  half  an  hour  ?   in  4  and  1  half  hours  ? 

d.  How  many  peck  baskets  will  it  take  to  hold  8  quarts  of 
plums  ?   32  quarts  ?    28  quarts  ? 

e.  How  many  times  must  a  quart  measure  be  filled  to  meas- 
ure out  a  bushel  and  a  half  of  cranberries  ? 

/.  How  many  quart  cans  will  it  take  to  hold  8  and  1  half 
gallons  of  milk  ? 

g.  How  many  times  must  you  pick  a  pint  dish  full  of  ber- 
ries to  fill  a  basket  that  holds  a  j)eck  ?    that  holds  a  peck  and 

2  quarts  ? 


MISCELLANEOUS  EXAMPLES,  65 

h.  How  many  pounds  of  beef  at  12  cents  a  pound  will  be 
required  to  pay  for  9  quarts  of  cherries  at  8  cents  a  quart  ? 

i.  If  $81  is  paid  for  9  weeks'  labor,  what  is  paid  for  4 
weeks'  labor  ?   for  7  weeks'  ?   for  12  weeks'  ? 

j.  If  6  yards  of  muslin  cost  60/,  what  will  7  yards  cost  ? 
10  yards?   12  yards  ? 

k.  If  7  persons  consume  a  barrel  of  sugar  in  9  months, 
how  long  will  it  last  1  person  ?  How  long  will  it  last  8  per- 
sons ? 

1.  If  a  family  of  3  persons  consume  a  barrel  of  flour  in 
6  months,  how  long  will  it  last  if  2  persons  are  added  to  the 
family  ? 

165.    Repeat  the  following  table  of 

Time. 

60  seconds  =  1  minute. 
60  minutes  =1  hour. 
24  hours     =  1  day. 
7  days       =1  week. 
52  weeks  and  1  day  \ 
or  365  days  ; 

100  years    =1  century. 

a.  How  many  seconds  are  there  in  half  a  minute  ? 

b.  How  many  minutes  are  there  in  half  an  hour  ?  in  a 
quarter  of  an  hour  ?  in  an  hour  and  a  quarter  ?  in  an  hour 
and  a  half  ? 

The  word  ''day"  as  used  in  the  table  denotes  the  interval  of  time  be- 
tween one  midnight  and  the  next.  As  used  in  the  following  example,  it 
denotes  the  interval  of  time  between  sunrise  and  sunset. 

c.  When  the  sun  rises  ^t  six  o'clock  and  sets  at  six  o'clock, 
how  long  is  the  day  ?   how  long  is  the  night  ? 

d.  When  the  days  are  11  hours  long,  how  long  are  the 
nights  ? 

e.  If  a  person  is  84  years  old,  how  many  years  longer  must 
he  live  to  be  a  century  old  ? 


= 1  year. 


66  DIVISION. 

There  are  12  months  in  the  year,  with  days  as  follows : 

**  Thirty  days  hath  September, 
April,  June,  and  November ; 
All  the  rest  have  thirty- one. 
Except  the  second  month  alone, 
To  which  we  twenty-eight  assign, 
Till  leap-year  gives  it  twenty-nine." 

/.    Name  the  months  that  have  30  days. 

g.  What  month  has  28  days  ?  How  many  days  has  it  in 
leap-year  ? 

h.  How  many  days  has  each  of  the  other  months  ? 

i.  The  winter  months  are  December^  January,  and  Feh- 
ruary.     How  many  days  are  there  in  winter  ? 

J.  The  spring  months  are  March,  April,  and  May.  How 
many  days  are  there  in  spring  ? 

k.  The  summer  months  are  June,  July,  and  August.  How 
many  days  are  there  in  summer  ? 

1.  The  autumn  months  are  September,  October,  and  No- 
vember.    How  many  days  are  there  in  autumn  ? 

166.   Repeat  the  following  tables : 

Numbers. 

16  ounces  =1  pound.  12  dozen  =  1  gross. 

2000  pounds  =  1  ton.  20  units  =1  score. 

12  gross  =  1  great  gross. 

222.  If  an  ounce  of  chocolate  is  used  at  each  meal,  for  how 
many  meals  will  2  pounds  of  chocolate  serve  ? 

22.  If  200  pounds  of  beef  are  put  into  a  barrel,  how  many 
barrels  will  it  take  to  hold  a  ton  ? 

o.  How  many  dozen  buttons  are  there  on  a  card  that  con- 
tains half  a  gross  ?     How  many  buttons  on  the  card  ? 

p.  The  allotted  age  of  man  being  "threescore  years  and 
ten,"  how  old  is  a  man  who  has  lived  17  years  beyond  the 
allotted  age  ? 


MISCELLANEOUS  EXAMPLES.  67 

167.   Miscellaneous  Examples  for  the  Slate. 

114.  A  farmer  has  45  sheep  and  18  lambs.  How  many  more 
sheep  has  he  than  lambs  ?     How  many  of  both  has  he  ? 

115.  A  flour-dealer  sold  28  barrels  of  flour  to  one  man,  33  to 
another,  17  to  a  third,  and  then  had  115  barrels  left.  How 
many  barrels  had  he  at  first  ? 

116.  If  6  sheets  of  paper  are  required  to  make  a  book  of  192 
pages,  how  many  pages  are  printed  on  one  sheet  ? 

117.  At  4/  a  dozen  for  hooks,  what  will  it  cost  for  hooks  for 
4  closets  having  18  hooks  in  each,  and  2  closets  having  30 
hooks  in  each. 

118.  Belle  gave  a  half-dollar,  a  dime,  and  3  cents  for  a 
fan,  and  had  ^2.bS  left.  How  much  had  she  at  first  ?  How 
many  cents  did  the  fan  cost  ? 

119.  A  newsboy  sold  16  Journals  at  3  ^  apiece,  23  Heralds 
at  2f  apiece,  13  Advertisers  at  4/^  apiece,  and  18  Transcripts 
at  4/  apiece.     What  did  he  receive  for  all  ? 

120.  At  $  2  a  pound  for  worsted,  what  will  1  ounce  cost  ? 

121.  Kate  bought  a  pound  and  a  quarter  of  worsted  at  $  2  a 
pound,  a  thimble  for  85  cents,  and  a  crochet-needle  for  7  cents. 
How  much  did  she  pay  for  all  ? 

122.  Mr.  Hubbard  bought  6  gross  of  lead-pencils  at  $1.80  a 
gross.     What  did  he  pay  for  the  lot  ? 

123.  If  1  gross  of  pencils  can  be  bought  for  $  1.80,  what  is 
that  per  dozen  ? 

124.  Here    are   the   charges   for                     charges. 
Olive's  bonnet.     What  did  it  cost  ?  ^^^^ ^  ^-^^ 

125.  Olive's   bonnet   cost    $1.89  ^^^^^^ ^-^^ 

more  than  Mary's.  What  did  Mary's  RMon 63 

cost?  Lace 38 

126.  What  was  the  cost  of  both  Frame  and  making  1.60 
bonnets,    and    2    scarfs    at    $1.62  Amount  i 
apiece  ? 

127.  If  196  pounds  of  flour  cost  $  9.80,  what  cost  50  pounds  ? 

128.  At  $  3.20  a  bushel  for  cranberries,  what  cost  9  quarts  ? 


68  MISCELLANEOUS  EXAMPLES. 

129.  I  sent  a  message  of  34  words  by  telegraph,  paying 
25  cents  for  the  first  10  words,  and  for  each  remaining  word 
2  cents.     What  did  the  message  cost  ? 

130.  If  a  horse-car  makes  12  trips  a  day,  and  takes  on  the 
average  24  passengers  at  5  cents  apiece,  how  much  money 
does  the  conductor  take  in  a  day  ? 

131.  Dr.  Lamb  caught  13  and  a  half  pounds  of  trout,  worth 
36  cents  a  pound.     What  was  the  lot  worth  ? 

132.  When  oysters  are  38  cents  a  quart  and  shad  12  cents  a 
pound,  what  shall  I  pay  for  a  pint  of  oysters  and  3  and  a  half 
pounds  of  shad  ? 

133.  If  a  horse-shoer  puts  8  nails  in  each  shoe  and  has  used 
up  1376  nails,  how  many  shoes  has  he  set  ?  How  many  horses 
has  he  shod,  each  horse  being  shod  all  round  ? 

134.  If  a  watch  loses  7  seconds  a  day,  how  many  seconds  will 
it  lose  in  a  year  ?     How  many  minutes  ? 

135.  At  4  cents  a  dozen  for  clothes-pins,  what  will  300 
clothes-pins  cost  ? 

136.  If  you  buy  6  papers  containing  25  needles  each,  for  25 
cents,  how  many  needles  do  you  get  for  1  cent  ? 

137.  At  62  cents  a  bushel,  what  will  5  and  a  half  bushels  of 
potatoes  cost  ? 

138.  A  clothier  bought  12  coats  for  $  13.92,  and  sold  them 
for  $  1.50  apiece.     What  did  he  gain  on  the  lot  ? 

139.  If  two  gallons  of  lemonade  are  made  with  14  lemons 
costing  36/  a  dozen,  and  2  pounds  of  sugar  costing  11/  a 
pound,  what  is  the  cost  of  a  gallon  of  lemonade  ?  of  a  quart  ? 
of  one  glass  holding  half  a  pint  ? 

140.  If  a  family  consume  2  quarts  of  milk  a  day  at  7  cents 
a  quart  from  October  1  to  April  1,  and  3  quarts  a  day  at  6  cents 
a  quart  from  April  1  to  October  1,  what  is  the  cost  of  milk  for 
the  year  ? 

141.  A  man  bought  350  cords  of  wood  for  $  962.50.  If  he 
sold  100  cords  at  $2.65  a  cord,  175  cords  at  $3.20  a  cord,  and 
the  rest  at  $  3.33  a  cord,  did  he  gain  or  lose,  and  how  much  ? 


MISCELLANEOUS  EXAMPLES. 


69 


142.  How  many  pins  are  there  in  a  paper  having  12  rows, 
and  30  pins  in  a  row  ?  If  you  can  buy  the  pins  for  18  cents, 
how  many  pins  do  you  get  for  1  cent  ? 

143.  Which  is  cheaper,  to  buy  360  pins  for  18  cents,  or  a 
paper  containing  12  rows  of  20  pins  each,  for  10  cents  ? 

144.  As  Mark  rode  to  his  uncle's,  he  watched  one  of  the 
wagon-wheels  and  found  it  turned  550  times.  The  wheel 
measured  13  feet  round.  How  many  feet  did  he  ride  ?  How 
many  miles  ? 

145.  Alfred  can  walk  around  a  certain  pond  in  9  minutes, 
and  Bertram  can  walk  around  in  8  minutes.  How  many  times 
can  Alfred  walk  around  while  Bertram  walks  around  36 
times  ? 

146.  How  many  feet  is  it  from  the  first  floor  of  a  house  to 
the  attic,  which  is  reached  by  2  flights  of  stairs,  the  first  having 
16  steps,  and  the  second  13  steps,  each  step  being  8  inches 
high,  and  12  inches  making  1  foot  ? 

147.  Here  is  a  bill  of  goods  bought  by  Mr.  James  Springer 
of  R.  H.  White  &  Co.  Find  what  ali  the  goods  cost,  and  make 
a  copy  of  the  bill  upon  paper. 

Boston,  Oct.  7,  1878. 
G^^l.  Ja??zed  (^/iu'??.^e^  ""^Otigllt  Of  E.  H.  WHITE  &  00. 


S  yu'U^  z^eAe^ @   ^  S.^{? 

/ 

^ 

4       ''        cadd77tei.e "             .^6^ 

£      -       ac'/d -         S.(y/9 

§  -^mae^                   .                            ''            .  S§ 

/  /^/,   ^4.^^;     1/  cap?,e,    //.i^<^ 

/  c/o-iieTt  /lac^  aocu^ 

7^ 

/ 

Received  'payment,                  h 

M.  s^if.  ^m. 

0 
i(-i^' 

\  ^o. 

70 


DRILL    TABLE. 


168.     DRILL  TABLE  No.  2. 


simple  Numbers. 

A 

B 

c 

D 

E 

F 

M 

41- 

N 

5  tu 

O 

viu  xy 

4 

16 

25 

39 

67 

66 

28 

987 

6909 

7 

19 

32 

43 

61 

67 

49 

996 

2985 

2 

22 

29 

38 

55 

61 

24 

872 

4360 

6 

14 

35 

46 

49 

66 

54 

327 

2943 

3 

20 

20, 

37 

64 

68 

36 

436 

2616 

8 

13 

34 

44 

56 

62 

72 

216 

1728 

5 

17 

30 

40 

60 

70 

46 

543 

1086 

7 

23 

27 

45 

58 

63 

56 

842 

5894 

11 

15 

33 

41 

52 

69 

88 

471 

1884 

9 

21 

31 

47 

59 

64 

27 

174 

1044 

10 

18 

28 

42 

53 

71 

66 

862 

3448 

12 

24 

36 

48 

60 

72 

96 

854 

1708 

G 

H 

1 

J 

K 

L 

81 

534 

3738 

74 

90 

99 

115 

124 

135 

16 

291 

2619 

80 

93 

97 

110' 

'121 

139 

39 

487 

3896 

73 

86 

98 

114 

126 

134 

68 

390 

1950 

82 

95 

103 

118 

130 

143 

94 

876 

2628 

75 

85 

101 

109 

128 

138 

63 

912 

5472 

79 

91 

108 

116 

122 

133 

42 

396 

2772 

81 

87 

107 

113 

131 

142 

33 

483 

1932 

76 

89 

100 

111 

125 

136 

96 

236 

1880 

78 

94 

106 

117 

123 

141 

57 

387 

1161 

83 

88 

104 

112 

127 

137 

18 

897 

4485 

77 

92 

102 

119 

129 

140 

48 

624 

5616 

'  84 

■96 

108 

120 

132 

144 

57 

246 

1476 

DRILL  EXERCISES. 


71 


169.     Exercises 

Examples  for  the  Slate. 

Multiply  N 

21.  By  2.  26.  By  7. 

22.  By  3.  27.  By  8. 

23.  By  4.  28.  By  9. 

24.  By  5.  ^a   By  11. 
^J.  By  6.  30.  By  12. 

Multiply  M 
31.  By  23.  .?^.  By  76. 

^£  By  54.  34.  By  98. 


Divide  N 

35.  By  2. 

4^. 

By  7. 

,56.  By  3. 

4i. 

By  8. 

57.   By  4. 

^^. 

By  9. 

38.  By  5. 

^. 

By  11. 

,5R   By  6. 

44. 

By  12. 

Divide  0 

45-  By  2. 

50. 

By  7. 

^5.  By  3. 

51. 

By  8. 

47.  By  4. 

52. 

By  9. 

4^.  By  5. 

53. 

By  11. 

49.  By  6. 

54. 

By  12. 

55.  Divide  0 

ty 

P. 

56.  Divide  0 

ty 

M. 

57.  Divide  0  by  N. 

Miscellaneous. 

58.  What  will  P  yards  of  cloth  cost 

at  M  cents  a  yard  ? 

59.  If  a  person  buys  goods  for  N 

cents,  and  gives  in  payment  a 


upon  the  Table. 

10-dollar  bill,  what  should  he 
receive  in  return  ? 

60.  At  P  cents  each,  how  many  pears 

can  you  buy  for  N  cents  ? 

61.  Divide  0  by  P  and  add  N  to  the 

quotient. 

62.  If  7  men  can  do  a  piece  of  work 

in  M  days,  in  how  many  days 
can  8  men  do  it  ? 

Oral  Practice. 

63.  Multiply  qhy  r  and  add  s  to  the 

product. 

64.  Multiply  r  by  s  and  add  t  to  the 

product. 

65.  Multiply  shj  t  and  add  ii,  to  the 

product. 

66.  Divide  A  and  B  each  by  2. 

67.  Divide  A,  B,  and  C  each  by  3. 

68.  Divide  B  to  D  each  by  4. 

69.  Divide  B  to  E  each  by  5. 

70.  Divide  B  to  F  each  by  6. 

71.  Divide  B  to  F  each  by  7;  by  8; 

by  9;  by  10;  by  11;  by  12. 

72.  Divide  G  by  7. 

73.  Divide  G  and  H  each  by  8. 

74.  Divide  G,  H,  and  I  each  by  9. 

75.  Divide  G  to  J  each  by  10. 

76.  Divide  G  to  K  each  by  11. 

77.  Divide  G  to  L  each  by  12. 

78.  Multiply  g  by  r  and  divide  the 

quotient  by  s. 


72  READING  AND    WRITING. 


SEOTIO]^    VI. 

Numbers  from  Ten  Thousand  to  Millions. 

170.  Count  by  tens  from  ten  to  a  hundred. 

Count  by  ten-thousands  from  ten  thousand  to  a  hundred 

thousand. 

Ten  thousand  is  written        10000  or  10,000. 

Twenty  thousand  is  written  20000  or  20,000. 

And  so  on. 

How   many   figures    are   needed   to   write   ten-thousands  ? 
Which  figure  shows  how  many  ten-thousands  there  are? 

171.  Read  the  following : 

a.   10,000.  c.    75,000.  e.   98,000.  g.  48,007. 

h.   40,000.  d.   17,000.  /.   36,549.  h.  93,080. 

Turn  to  page  77,  and  read  the  numbers  written  in  column  C. 

172.  Write  in  figures  the  numbers  made  up  of  ten-thousands 
from  ten  thousand  to  ninety  thousand. 

Write  in  figures  the  following : 

1.  Thirty-four  thousand,  two  hundred  seventy-five. 

2.  Eighty-nine  thousand,  thirty-six. 

3.  Fifty-four  thousand,  five  hundred  two. 

4.  Fifteen  thousand,  twelve. 

Let  the  teacher  dictate  other  numhers  for  the  pupil  to  write. 

173.  Count  by  hundred-thousands  from  one  hundred  thou- 
sand to  ten  hundred  thousand. 

One  hundred  thousand  is  written  100000  or  100,000. 

Two  hundred  thousand  is  written  200000  or  200,000. 

And  so  on. 

How  many  figures  are  needed  to  write  hundred-thousands  ? 
Which  figure  shows  how  many  hundred-thousands  there  are  ? 


SIMPLE  NUMBERS.  73 

174.  Eead  the  following : 

i.    400,000.        k.  527,810.         m.  324,517.  o.   407,086. 

j\   438,261.        1.    906,224.         n.    872,239.         p.  370,805. 

Turn  to  page  77,  and  read  the  numbers  written  in  column  B, 

175.  Write  in  figures : 

5.  Five  hundred  sixty-eight  thousand,  four  hundred. 

6.  Four  hundred  twenty  thousand,  twenty-six. 

7.  Nine  hundred  nine  thousand,  nine. 

8.  Eight  hundred  forty-one  thousand,  live  hundred  three. 

Let  the  teacher  dictate  other  numbers  for  the  pupil  to  write. 

176.  The  figures  in  the  fourth,  fifth,  and  sixth  places  taken 
together  form  a  group  called  the  thousands'  group,  while 
the  figures  in  the  first,  second,  and  third  places  form  a  group 
called  the  units'  group.  The  figures  in  the  units'  group 
express  units,  tens,  and  hundreds  of  units,  while  the  figures 
in  the  thousands'  group  express  units,  tens,  and  hundreds 
of  thousands.  In  writing  numbers,  separate  the  groups  of 
figures  hy  a  comma,  as  shown  in  Articles  173  and  174. 

177.  Beyond  the  second  group  are  higher  groups.  A 
third  group  of  figures,  expressing  units,  tens,  and  hundreds 
of  millions,  is  shown  in  the  following 

TABLE. 


Sfl53       Sflo  SS-^ 

^    ^    ^        ^    ^    .d  ^    ^    -^ 

HJ-4J-t-5  ^J^J.^J  rS'^OT 

OOOt^  OOrt<  COCNrH 

7  2   5,  ,   3   2   8  ,  7  8  3^  Figures. 

3d  group.         2d  group.  ist  group. 

Millions.    Thousands.  Units. 

How  many  figures  are  needed  to  write  millions?   ten-mil- 
lions ?   hundred-millions  ? 


74  SIMPLE  NUMBERS. 

Which  figure  shows  how  many  millions  there  are  ?  which 
shows  how  many  ten-millions  there  are  ?  which  shows  how 
many  hundred-millions  there  are  ? 

The  number  written  in  the  table  is  read 

"  Seven  hundred  twenty-five  million,  three  hundred  twenty- 
eight  thousand,  seven  hundred  eighty-three." 

178.  Kepeat  the  following  table  : 

Ten  ones  (or  units)  make  a  ten. 

Ten  tens  make  a  hundred. 

Ten  hundreds  make  a  thousand. 

Ten  thousands  make  a  ten-thousand. 

Ten  ten-thousands  make  a  hundred-thousand. 

Ten  hundred-thousands  make  a  million. 

179.  Eead  the  following  : 

q.  2,684,500.  t.   8,109,019.  w,  1,326,709. 

r.   9,275,405.  u.   4,414,8.93.  x,   5,070,890. 

s.  3,118,184.  V.    6,005,928.  y.   7,654,321. 

180.  Turn  to  page  76,  and  write  in  figures  the  numbers 
given  in  column  A. 

Let  the  teacher  dictate  other  numbers  to  millions  for  the  pupil  to  write. 

181.     Miscellaneous  Examples  for  the  Slate. 

9.  How  much  must  be  paid  for  625  acres  of  land,  at  $  175 
an  acre  ? 

10.  It  is  about  25,000  miles  round  the  earth.  How  many 
days  would  it  take  a  person  to  travel  this  distance  if  he  trav- 
elled 40  miles  a  day  ? 

11.  Mr.  Gaines  owned  real  estate  valued  at  $56,000.  Of 
this  he  sold  land  worth  $5,000  and  a  house  worth  $6,750. 
What  was  the  value  of  what  he  had  left  ? 

12.  At  7  cents  a  pound,  how  many  cents  will  a  dealer  receive 


MISCELLANEOUS  EXAMPLES.  75 

for  50  boxes  of  soap  each  containing  72  pounds  ?     How  many 
dollars  will  he  receive  ? 

13.  How  many  square  miles  are  there  in  the  United  States, 
the  Northern  Lake  region  containing  484,339  square  miles; 
the  Atlantic  Slope,  304,530  square  miles;  the  Gulf  region, 
1,683,303  square  miles ;  and  the  Pacific  Slope,  854,314  square 
miles  ? 

14.  In  1865  Massachusetts  had  1,267,031  inhabitants,  and  in 
1875  she  had  1,651,912  inhabitants.  What  was  the  increase 
in  ten  years  ? 

15.  How  many  seconds  are  there  in  the  month  of  January  ? 

16.  When  Charles  is  11  years  old,  how  many  hours  has  he 
iived,  if  in  that  time  there  have  been  3  leap-years  ? 

17.  The  great  bell  at  Moscow  weighs  448,000  pounds.  How 
many  tons  does  it  weigh  ? 

18.  In  one  part  of  her  orbit  the  moon  is  224,000  miles  from 
us.  If  a  cannon-ball  moves  at  the  rate  of  16  miles  a  minute, 
in  how  many  minutes  will  it  move  through  this  distance  ? 

19.  In  2,468  cocks  of  hay,  averaging  78  pounds  each,  how 
many  pounds  ?     How  many  tons  ? 

20.  A  man  bought  42  bushels  of  potatoes  for  $7.50,  and 
sold  them  for  80  cents  a  peck.  How  much  did  he  receive  for 
them  ?  •  How  much  did  he  gain  ? 

21.  At  4  cents  apiece,  what  will  it  cost  for  slats  to  fence  a 
lot  of  land  having  4  sides  each  250  feet  long,  5  slats  being 
required  for  each  foot  of  length  ? 

22.  What  will  be  the  cost  of  ties  at  35  cents  each,  to  build  a 
mile  of  railroad,  there  being  6  ties  to  a  rod  and  320  rods  in  a  mile  ? 

23.  A  farmer  bought  5,960  feet  of  boards  at  %  0.03  a  foot, 
and  gave  in  payment  cheese  at  $  0.12  a  pound.  How  many 
pounds  did  it  take  ? 

24.  How  many  are  3,687,543  +  245,871  +  3,684  +  932,185  ? 

25.  From  4,357,859  +  248,946  take  2,783,947. 

26.  Multiply  7,285  by  394  and  by  207. 

27.  Divide  4,893,683  by  14  and  by  25. 


76  DRILL    TABLE. 

182.     DRILIi   TABLE   No.  3. 
Simple  Numbers. 
A 

Five  million^  eighty-one  thousand,  two  hundred  six. 

One  million,  five  hundred  seven  thousand,  forty-one. 

Eight  million,  three  hundred  twenty-one  thousand,  forty. 

Five  million,  twenty-nine  thousand,  three  hundred. 

Four  million,  seven  hundred  thousand,  four  hundred  four. 

Three  million,  forty  thousand,  two  hundred  thirty-four. 

Eight  million,  six  hundred  ten  thousand,  ninety-one. 

Nine  million,  fLYe>  hundred  thirty  thousand,  seven. 

Six  million,  one  hundred  sixteen  thousand,  six  hundred. 

Two  million,  seventy-nine  thousand,  one  hundred  eighty. 

Five  million,  eighty-four  thousand,  four  hundred  nine. 

Three  million,  nine  hundred  thousand,  one  hundred  two. 

Eight  million,  seven  hundred  six  thousand,  fifty-nine. 

One  million,  forty-eight  thousand,  two  hundred  eight. 

Nine  million,  eight  thousand,  three  hundred  twenty. 

Four  million,  twenty-three  thousand,  five. 

Five  million,  seven  hundred  sixty-eight. 

Eight  million,  seventy-two  thousand,  eighty-nine. 

Three  million,  five  hundred  seven  thousand. 

Two  million,  three  hundred  forty-eight  thousand,  eight. 

Nine  million,  twelve  thousand,  six  hundred  ninety. 

Seven  million,  seven  thousand,  seven  hundred. 

Six  million,  fifty  thousand,  two  hundred  ninety-one. 

Four  million,  five  hundred  eighty-eight. 

Seven  million,  seven  hundred  thousand,  four  hundred  six. 


DRILL  EXEEGISES, 


77 


DRILL  TABLE  No.  3 

{continued). 


B 

987  449 
905  788 
679  435 
369  663 
153  967 
^^%  455 
806  399 
917  517 
927  931 
953  205 
667  871 
728  431 
845  219 
144  SQ>S 
225  189 
199  598 
941  529 
795  721 
734  256 
323  583 
824  174 
769  416 
108  824 
872  892 
444  764 


c_ 

2  5~3i7 

4  5  995 

3  7  872 

8  8  324 

7  5  436 

5  7  216 

9  4  503 

6  6  842 
19  471 

6  7  174 
2  6  862 
9  6  354 

2  9  534 
16  111 
5  4  197 

4  5  490 

8  0  876 

3  4  902 

7  6  396 

5  6  484 

4  9  235 

7  4  387 

6  2  897 
3  8  624 

8  7  476 


183.  Exercises  upon  the  Tables 

Addition. 

79.  Write  A  in  figures. 

80.  Add  A,  B,  and  C. 

81.  Add  in  A  from  1  to  5;  2  to  6,  etc. 

82.  Add  in  B  from  1  to  8;  2  to  9,  etc. 
8S.  Add  in  C  from  1  to  11 ;  2  to  12, 

etc. 

Subtraction. 

84.  From  A  take  B. 

85.  From  B  take  C. 

86.  From  A  take  C. 

87.  From  1,001,001  take  B. 

88.  From  9,900,600  take  A. 

Multiplication. 

Multiply  B 

89.  By  2.       92.  By  5.       95.  By  8. 

90.  By  3.       93.  By  6.       96.  By  9. 

91.  By  4.       94.  By  7.       97.  By  10. 

Multiply  D 

98.  By  12.    101.  By  67.  IO4.  By  168. 

99.  By  34.    102.  By  78.  105.  By  312. 
100.  By  57.    103.  By  87.  106.  By  970. 

Division. 

Divide  A 

107.  By  2.      110.  By  5.     113.  By  8. 

108.  By  3.      111.  By  6.     II4.  By  9. 

109.  By  4.      112.  By  7.     ii<5.  By  11 

Divide  B 
116.  By  17.  119.  By  904. 

i27.  By  45.  i^(?.  By  686. 

118.  By  198.  '    i^i.  ByC. 

122.  Divide  A  by  B. 

2;^^.  Divide  A  by  C. 


"^S  FACTORS. 

SEOTIOIsr   YII. 

FACTORS. 

184.  What  numbers  multiplied  together  will  make  6  ? 
Ans.  2  and  3,  also  1  and  6 ;  thus,  2x3  =  6,  and  1x6  =  6. 

A  number  that  may  be  used  as  multiplicand  or  multi- 
plier to  make  another  number  is  a  factor  of  that  number. 
Thus,  2  and  3  are  factors  of  6 ;  so  also  are  1  and  6. 

a.  Name  two  factors  of  4  ;  of  9  ;  of  12  :  of  15. 

185.  iSTame  some  factors  of  8  besides  8  and  1. 

A  number  that  has  other  factors  besides  itself  and  1  is  a 
composite  number.     Thus,  8  is  a  composite  number. 

186.  Name  the  factors  of  5. 

A  number  that  has  no  other  factors  besides  itself  and  1 
is  a  prime  number.     Thus,  5  is  a  prime  number. 

187.    Exercises. 

b.  Which  of  the  following  numbers  are  prime  and  which  are 
composite:  6?   7?   8?   10?  11?  12?   13?   14? 

c.  Write  the  composite  numbers  from  1  to  30 ;  from  30  to  50. 

d.  Write  the  prime  numbers  from  1  to  30 ;   from  30  to  50. 

188.  e.  Name  any  factors  of  12  that  are  prime  numbers. 
Name  any  that  are  not  prime. 

A  factor  that  is  a  prime  number  is  a  prime  factor. 
Thus,  2  and  3  are  prime  factors  of  12. 

189.  A  composite  number  equals  the  product  of  all  its 
prime  factors.     Thus,  12  =  2x2x3. 


FACTORS.  79 

190.    Exercises. 

Separate  the  following  numbers  into  their  prime  factors  and 
write  the  results,  thus :  6  =  2x3;   8  =  2x2x2,  and  so  on. 

/.   6.  h.   10.  j.   18.  1,   2S.  22.   40. 

g.  8.  i.    16.  k.  20.  222.  30.       ^     o.  45. 

191.  p.  Write  all  the  prime  factors  of  18  and  of  24,  and 
name  the  numbers  that  are  factors  of  both. 

.       (  The  prime  factors  of  18  are  2,  3,  and  3« 
(  The  prime  factors  of  24  are  2,  2,  2,  and  3, 
The  factors  of  both  are  2,  3,  and  their  ]3roduct  6. 

192.  A  number  that  is  a  factor  of  two  or  more  numbers 
is  a  common  factor  of  those  numbers.  Thus,  2,  3,  and  6 
are  common  factors  of  18  and  24. 

q.  Which  of  the  numbers  2,  3,  and  6  is  the  greatest  com- 
mon factor  of  18  and  24  ? 

193.    Exercises. 

r.   Name  a  common  factor  of  8  and  12 ;  of  10  and  15. 

s,  Name  all  the  common  factors  of  18  and  12. 

t.  ISfame  the  greatest  comm.on  factor  oi  18  and  12. 

What  is  the  greatest  common  factor 

H.  Of  9  and  12  ?        w.  Of  8  and  12  ?         j^.  Of  6  and  18  ? 

V.  Of  12  and  15  ?      x.  Of  18  and  20  ?       z.    Of  24  and  36  ? 

Multiples. 

194.  a.  Name  some  numbers  which  are  made  by  using  2 
as  a  factor.  Ans.  2,  4,  6,  8,  etc. 

A  number  made  by  using  another  number  as  a  factor  is 
a  multiple  of  the  number  so  used.  Thus,  tlie  numbers  2, 
4,  6,  and  8  are  multiples  of  2. 


80  FACTORS. 

195.    b.  Write  the  multiples  of  3  and  of  4  to  24. 

^^^^   f  Multiples  of  3  are  3,  6,  9,  12,  15,  18,  21,  24. 
'^^*  t  Multiples  of  4  are  4,  8,       12,  16,  20,        24. 

c.  Which  of  these  numbers  are  multiples  of  both  3  and  4  ? 

A  number  that  is  a  multiple  of  two  or  more  numbers  is 
a  common  multiple  of  those  numbers.  Thus,  12  is  a 
common  multiple  of  3  and  4 ;  so  also  is  24. 

d.  Which  of  these  numbers,  12  and  24,  is  the  least  common 
multiple  of  3  and  4  ? 

196.    Exercises. 

e.  Write  all  the  multiples  of  2  and  of  3  to  30. 

/.  Which  of  the  numbers  3'ou  have  written  are  common 
multiples  of  2  and  3  ? 

g.    Which  is  the  least  common  multiple  of  2  and  3  ? 

h.  Write  all  the  multiples  of  4  and  of  5  to  40. 

i.  Which  of  the  numbers  you  have  written  are  common 
multiples  of  4  and  5  ? 

j.    Kame  the  least  common  multiple  of  4  and  5. 

Name  any  common  multiple 

k.  Of  3  and  4 ;   of  3  and  9  ;    of  2,  3,  and  4. 
^      i.    Of  2  and  5  ;    of  4  and  9  ;    of  5,  10,  and  15. 

m.  Of  5  and  7 ;   of  8  and  12 ;   of  5,  8,  and  10. 

Name  the  least  common  multiple 

22.   Of  3  and  5.  s.    Of  4,  6,  and  8. 

0.   Of  6  and  7.  t.    Of  4,  8,  and  10. 

p.  Of  4  and  6.  u.   Of  4,  5,  and  6. 

(?.    Of  6  and  10.  v.   Of  3,  8,  and  10. 

r.    Of  9  and  12.  w.  Of  6,  9,  and  12. 

For  fuller  treatment  of  Greatest  Common  Factor  and  Least  Common 
Multiple,  see  Appendix,  x^ages  139,  140. 


COMMON  FRACTIONS. 


SECTION    YIII. 


COMMON    FRACTIONS. 

197.  Eichard  cut  a  melon  into  two  equal  parts  and 

gave  his  brother  one 
of  the  parts.  What 
part  of  the  melon 
did  each  then  have  ? 
(Art.  142.) 

a.  How  many  halves 
equal  a  whole  thing  ? 

198.  Three  boys  divided   a  pineapple  equally  among 
themselves.  What 
part  of  the  pineapple 
did  each  boy  have  ? 

Z).  How  many  thirds 
equal  a  whole  thing  ? 

c.  If     any     single  : 
thing  or  unit  is  divided 
into  four  equal  parts,  what  is  each  of  the  parts  called  ?    What 
are  two  of  the  parts  called  ?    three  of  the  parts  ? 

d.  If  a  unit  is  divided  into  five  equal  parts,  what  is  each  of 
the  parts  called  ?  two  of  the  parts  ?  three  of  the  parts  ?  four 
of  the  parts  ? 

e.  If  a  unit  is  divided  into  six  equal  parts,  what  is  each  of 
the  parts  called  ?   two  of  the  parts  ?   three  of  the  parts  ? 

/.    In  one  how  many  fourths  ?    thirds  ?    fifths  ?    sixths  ? 

199.  The  equal  parts  of  a  unit,  as  halves,  thirds,  fifths, 
etc.,  are  fractions.  Numbers  that  are  not  fractions  are 
.called  whole  numbers  or  integers. 


82  COMMON  FRACTIONS, 

g.  To  obtain  two  thirds,  into  how  many  equal  parts  is  the 
unit  divided  ?     How  man}^  parts  are  taken  ? 

200.  The  number  of  equal  parts  into  which  a  unit  is 
divided  is  the  denominator  of  the  fraction. 

Thus,  in  the  fraction  two  thirds,  the  denominator  is  three. 

201.  The  number  of  equal  parts  taken  is  the  numerator 
of  the  fraction. 

Thus,  in  the  fraction  two  thirds,  the  numerator  is  two. 

202.  The  numerator  and  denominator  are  called  the 
terms  of  the  fraction. 

h.   What  is  the  denominator  of  the  fraction  three  fifths  ? 
i.    What  is  the  numerator  ? 

Writing  Common  Fractions. 

203.  Two  thirds  is  written  as  in  the  margin,  the 
Numerator  3  nuuicrator  two  abovc  the  line,  and  the  de- 
Denominator..^        nominator  th^e  below. 

Exercises. 

204.  Write  in  figures  the  following : 

a.  One  half.  d.   Three  fourths. 

b.  One  third.  e.   Four  fifths ;   five  fifths. 

c.  Two  thirds.  /.    Seven  twelfths ;  seven  fifths. 
g.  Write  any  fraction  you  please  having  for  a  denominator 

six ;    seven  ;    nine  ;    eleven. 

h.  Write  any  fraction  you  please  having  for  a  numerator 
four ;    six ;    seven  ;    nine  ;    twelve. 

205.  Kead  the  following,  and  tell  how  many  parts  the  unit 
is  divided  into,  and  how  many  parts  are  taken  in  each  case : 

i.   fapp^e.         k.   tday.         m.  $  A-         o.    f.  q.   H- 

J.   f  pear.  1.    J  quart.      n.    $  |.  p.    f .  r.    -j- J. 


COMMON  FRACTIONS. 


83 


To  change  a  Fraction  to  smaller  or  larger  Terms. 

206.  Mary  cut  an  apple  into  halves,  and  then  cut  one 

of  the  halves  into  two 
equal  parts.  What  part  of 
the  whole  apple  was  one 
of  these  smaller  parts  ? 

If  Mary  should  put 
these  two  fourths  to- 
gether, what  pg.rt  of  the 
whole  apple  would  they 
make? 

a.  One  half  eqiials  how  many  fourths  ? 

b.  Two  fourths  equal  how  many  halves  ? 

c.  If  a  piece  of  paper  one  inch  long  be 
cut  into  thirds,  and  each  of  these  thirds  be 
cut  into  two  equal  parts,  how  many  equal 
parts  will  the  whole  piece  be  cut  into  ? 

d.  What  part  of  an  inch  will  one  of  these 
smaller  parts  be  ? 

e.    One  third  equals  how  many  sixths  ? 
/.    Two  sixths  equal  how  many  thirds  ? 

207.  Compare  the  terms  of  the  equal  fractions  \  and  |. 
The  terms  of  the  fraction  ^  are  one  half  as  large  as  the 


i   i 


i 


terms  of  the  fraction  |. 


And  the  terms  of  the  fraction  | 


are  twice  as  large  as  the  terms  of  the  fraction  ^.     So, 

208.  If  both  terms  of  a  fraction  he  multiplied  hy  the  same 
number,  the  value  of  the  fraction  will  not  be  changed.     And, 

If  both  terms  of  a  fraction  be  divided  by  the  same  num- 
ber, the  value  of  the  fraction  will  not  be  changed. 

g.  By  what  will  you  multiply  the  terms  of  the  fraction  J  to 
change  it  to  fourths  ?    to  sixths  ?   to  eighths  ? 


S4  COMMON  FRACTIONS, 

h.  By  what  will  you  divide  the  terms  of  the  fraction  |  to 
change  it  to  J  ? 

i.  By  what  will  you  divide  the  terms  of  the  following  frac- 
tions to  change  them  to  halves  :    f  ?    j\?    ^^  ?    ^^  ? 

209.     Oral  Exercises. 

First  do  the  following  examples,  writing  the  answers 
upon  the  slate;  then  practise  doing  them  mentally  till 
you  can  name  the  results  at  sight. 

a.  Change  to  smaller  terms  :  I ;   f  ;   f  ;   f;  f  ;    f ;  A  ;    ^  ; 

b.  Change  to  smaller  terms:    ^^',  ^%',   ^^',    -j:'^;    ig;    ^s^  ^ 

c.  Change  to  smaller  terms:    if  5   M;   II;   A;    U  5    M; 

•f*;  i§;  if.  M;  H;  If;  M- 

c?.  Change  ^  to  fourths  ;  to  sixths  ;  to  tenths  ;  to  twelfths. 
e.  Change  -^  to  sixths ;  to  ninths ;  to  twelfths  ;  to  fifteenths. 
/.  Change  f  to  sixths  ;  to  ninths ;  to  twelfths ;  to  fifteenths. 
g.    Change  to  twenty-fourths :   i;    J-;    §;    J;    f;    J;    f;    J; 

I ;  t ;  i ;  tV  ;  tV ;  t\* 

iz.    Change  to  thirtieths :    J;    J;    f;   J;   f;    J;    f;    f;   ^i^ ; 

To ;  tV  ;  A" ;  t^s  ;  A ;  A* 

To  change  a  Fraction  to  its  smallest  Terms. 

210.  To  change  a  fraction  to  its  smallest  terms,  Divide  hj 
all  the  factors  that  are  common  to  numerator  and  denomi- 
nator ;  or  divide  hoth  terms  hy  their  greatest  common  factor. 

211.    Examples  for  the  Slate. 

Change  the  following  to  equivalent  fractions  of  smallest  terms : 
a.  Do  the  work  mentally,  and  write  the  result  as  it  is 

if  ~  f  •        written  in  a. 

(1.)  M-  (4.)  li-  •  (7.)  if. 
(2.)  if.  (5.)  n.  (8.)  U- 
(3.)  ^.        (6.)  H.        (9.)  If. 

For  other  examples,  see  j)age  129. 


(10.)  |§. 

(13.)  n- 

(11.)  fg. 

(14.)  IS. 

(12.)  U- 

(15.)    T*/^, 

COMMON  FRACTIONS. 


85 


To  change  Improper  Fractions  to  Whole  or  to  Mixed  Numbers. 

212.  A  fraction  whose  numerator  equals  or  exceeds  its 
denominator  is  called  an  improper  fraction. 
Thus,  I,  j,  and  |  are  called  improper  fractions, 
a.    4  half  dollars  equal  how  many  dollars?      Z).    J  apples 
,1^  ^^^        equal    how    many    whole 

:^^^     apples  ? 


(&.)  Since  2  halves  make 

2)  7       a  whole  one,  |-  apples 

3I     will    equal    as    many 

whole  apples  as  there 

are  2's  in  7,  which  is  3,  and 

1  half  remains.  ^^5.  3^  apples. 

213.  The  number  3J 
consists  of  a  whole  number  and  a  fraction.  A  number  consist- 
ing of  a  whole  number  and  a  fraction  is  a  mixed  number. 

214.    Oral  Exercises. 

c.  Alvin  picked  berries  into  a  dish  that  held  J-  of  a  quart  j 
if  he  filled  the  dish  8  times,  how  many  quarts  did  he  pick  ? 

d.  How  many  quarts  are  there  in  f  quarts  ?    in  f  ?  in  -y-  ? 

e.  How  many  dollars  are  there  in  $  -\0-  ?    in  $  V"  ^   in  $  -^  ^ 
/.    By  what  do  you  divide  to  change  halves  to  units  ?    to 

change  thirds  ?    fourths  ?    fifths  ?    sevenths  ?    tenths  ? 

^.    Change  to  whole  numbers :   | ;    | ;    -V"  j    t  j   V"  ?    §  5   ¥-• 
iz.   Change  to  mixed  numbers  :   J ;    | ;  -V^- ;   J ;    J ;  J^- ;    ^- ; 

i.    Change  to  whole  or  mixed  numbers:   |;  -^;  f ;  -^ ;  ^-] 

¥;  ¥;  -V-;  ¥;  V-;  ¥-;  ¥-;  ¥;  -¥;  ¥;  ¥-5  M- 

215.    Examples  for  the  Slate. 
Change  the  following  fractions  to  whole  or  mixed  numbers, 
writing  the  results  as  in  a  and  b,  below. 

a.  V-  =  16|.     (16.)1F.     (18.)iF.     (20.)iF.     (22.)  W- 
Z>.  ^  =  19J.     (17.)  ^|2.     (19.)  ^s..     (21.)  if  1.     (23.)  W- 


86 


COMMON  FRACTIONS. 


To  change  a  "Whole  or  a  Mixed  Number  to  an  Improper  Fraction. 
216.    a.  Two  dollars  equal  how  many  half  dollars  ? 

^^  h.  How  many  half  apples 

are  there  in  3  apples  ?    in 


4  ?    in  5  ?    in  7  ? 

c.  By  what  do  you  mul- 
tiply to  change  units  to 
halves  ? 

d.  How  many  half  apples 
are  there  in  3 J  apples  ? 

Since  in  1  there  are  2  halves, 
{d.)        m  3  there  are  3  times  2  halves,  or  6  halves,  which  with 
^i  =  2-       1  half  added  are  7  halves.     Ans.  ^  apples. 

217.    Oral  Exercises. 

e.  Ada  had  2  yards  of  ribbon,  which  she  made  into  knots 
of  1  third  of  a  yard  each.     How  many  knots  did  she  make  ? 

/.    How  many  thirds  are  there  in  2  ?    in  3  ?    in  5  ?    in  7  ? 

g*.    By  what  do  you  multiply  to  change  a  number  to  thirds  ? 

h.  How  many  thirds  are  there  in  4§?    in  5J  ?    in  6f  ? 

i.  If  it  takes  1  fourth  of  a  3^ard  of  cloth  to  make  a  cap,  how 
many  caps  can  be  made  from  5  j^ards  ?   from  5 J  yards  ? 

J.  By  what  do  you  multiply  to  change  a  number  to  fourths  ? 
fifths?    sixths?    eighths?    ninths?    tenths? 

k.  Change  3  to  fourths  ;  4  to  fifths  ;  5  to  eighths ;  8  to  tenths. 

I.    Change  to  improper  fractions:  IJ;  8J;  2J;  4J;  7 J ;  3J. 

m.  Change  to  improper  fractions :  If ;  4J ;  8| ;  3f  ;  S^q  ; 
8|;  6|;  8i;   5^;  7J ;  12^;  6^ ;   of;   2J;  4f;  3f;   Sf;  7|. 

218.    Examples  for  the  Slate. 

Change  the  following  numbers  to  improper  fractions,  writing 
the  result  as  in  a,  below. 


a.  16f=Aj4. 

(26.)  33J. 

(29.)  32f. 

(32.)  1211. 

(24.)  37^. 

(27.)  66$. 

(30.)  272i. 

(33.)  271  f. 

(25.)  17i. 

(28.)  301. 

(31.)  187f 

(34.)  368|. 

ADDITION  OF  FRACTIONS.  87 

ADDITION   OF   FRACTIONS. 
To  add  Fractions  having  a  Common  Denominator. 

219.  a.  In  J  of  a  day,  |  of  a  day,  and  f  of  a  day,  how  many 
fourths  ?     How  many  days  ? 

The  fractions  \,  |,  and  |  have  the  same  denominator,  4. 
Fractions  which  liave  the  same  denominator  are  said  to 
have  a  common  denominator. 

h.   In  adding  \,  |,  and  f,  which  terms  did  you  add  ? 

220.    Oral  Examples. 

c.  Charles  picked  ^  of  a  peck  of  berries,  William  picked  f 
of  a  peck,  and  Alfred  picked  ^  of  a  peck.  How  many  eighths 
did  all  pick  ?     How  many  pecks  ? 

d.  How  many  sevenths  are  f ,  f ,  and  \  ? 

e.  How  many  sixths  are  J,  f ,  and  f  ? 

/.    How  many  elevenths  are  ^j,  -j^,  and  y\  ? 

g.    Add  t\,  /_  and  \\,  i.   Add  ^,  ^^r,  and  /^. 

iz.  Add  ^0?  hh  <^nd  ^J.  j.   Add  ^^,  ^%,  and  ^. 

To  add  Fractions  not  having  a  Common  Denominator. 

221.  a.  James  worked  |  of  a  day  for  Mr.  Smith  and  J-  of 
a  day  for  Mr.  Clark.  How  many  fourths  of  a  day  did  he  work 
for  both  ?     How  many  days  ? 

b.  How  many  eighths  in  ^  plus  J  ?    in  J  plus  |  ? 

c.  In  I  plus  f  how  many  ninths  ?     How  many  units  ? 
cf.   In  f  plus  y^^  how  many  tenths  ?     How  many  units  ? 

222.  e.    Add  J,  |,  and  #. 

WRITTEN  WORK.  To  add  these  fractions,  they  must  be  changed 

i  =  ^  to   equivalent  fractions  having  a  common  de- 

f  -  "ft"  nominator. 

I  =  ^^  The   most    convenient    denominator   is   the 

TT     ^  -  -  least   common    multiple   of   all    the    denomi- 

^  ^        ^  -*  nators. 


88  COMMON  FRACTIONS. 

The  least  common  multiple  of  3  and  4  is  3  times  4,  or  12,  which  is 
also  a  multiple  of  6  * 

To  change  \  to  twelfths,  the  denominator  3  must  be  multiplied  by 
4.  Hence  the  numerator  must  be  multiplied  by  .4.  Thus,  \  is  found 
to  equal  ■^. 

In  a  similar  way  |  is  found  to  equal  ^%,  and  |  to  equal  l^.  Adding 
these  fractions,  we  have  ||,  or  1^-^,  for  the  sum.     Ans.  l\^. 

223.    Oral  Examples. 

/.  Add  i  and  \  ;  ^  and  J  ;  J-  and  J  ;  \  and  \  ;  J  and  ^^ ; 
J  and  ^ ;    ^  and  ^J^j ;    \  and  ^ ;    |  and  ■^. 

g.   Add  4^  and  ^ ;    J  and  i ;    J  and  i ;    ^  and  J ;    J-  and  ^. 

A.  Add  ^  and  f  ;    J  and  f  ;    f  and  f  ;   f  and  J  ;    f  and  f . 

i.  Add  J,  1,  and  J  ;  i  J,  and  i ;  ^,  \,  and  -jio  ;  J.  h  and 
T^;   i?  i,  and  ■^^. 

j.  Add  I,  J,  and  t ;  if,  and  J  ;  |,  f ,  and  t\ ;  |_^  ..|^  and 
T^;   f ,  f ,  and^y^. 

k.  Miss  Prencli  cut  from  a  piece  of  ribbon  §  and  |  of  a  yard, 
and  then  had  |  of  a  yard  left.     How  long  was  the  piece  ? 

224.    Examples  for  the  Slate. 

In  doing  the  following  examples,  write  the  work  as  in  e. 
Art.  222. 

35.  Add  f ,  /o,  and  ^%.  39.  Add  ig,  J,  and  yV 

36.  Acjd  ^^,  T^,  and  ^^,  40.  Add  -^,  2%,  and  f . 

37.  Add  tV,  f,t  and  f .  41.  Add  ^^2,  M.  and  ^V 

38.  Add  ^\,  f,  and  \.  42.  Add  yV,  %,  and  /^. 

In  doing  the  following  examples,  add  the  wdiole  numbers 
and  the  fractions  separately. 

43.    Add  5i,  4f,  and  7^.  44.    Add  4|,  9§,  and  18f . 

*  If  in  any  example  the  least  common  multiple  is  not  readily  seen,  the 
pupil  may  take  for  a  common  denominator  any  common  multiple  of  the 
denominators.  The  product  of  all  the  denominators  is  frequently  the  most 
convenient. 

t  What  should  first  be  done  to  this  fraction  ? 


SUBTRACTION  OF  FRACTIONS.  89 

SUBTRACTION    OF    FRACTIONS. 
To  subtract  Fractions  having  a  Common  Denominator. 

225.  Oral  Examples. 

a.  Jolin  owned  J  of  a  football,  and  sold  f  to  Burt.  JEIow 
many  eighths  did  he  own  then  ? 

Z).    I  -  §  -  what  ?  d.   From  -^^  take  ^^, 

c.    f  - 1  -  what  ?  e.   From  \^  take  yV 

How  do  you  subtract  when  the  minuend  and  subtrahend 
have  a  common  denominator  ? 

/.    Charles  had  a  melon,  and  gave  §  of  it  away.     What  part 
had  he  left  ? 
5-.  l-§?  1-1?  1-A?  1-A?  1-*?  2-J? 
h.  4-2- f?   4-2f?   12-3t7^?  20 -6§?  100 -12^? 

To  subtract  Fractions  not  having  a  Common  Denominator. 

226.  Oral  Examples. 

J.  Silas  could  have  the  use  of  a  boat  for  half  an  hour.  After 
he  had  used  it  a  quarter  of  an  hour^  how  much  longer  could  he 
use  it  ? 

j.  If  1^  of  a  yard  of  muslin  be  cut  from  |  of  a  yard,  how 
many  eighths  will  be  left? 

k.  J  less  y'^j  are  how  many  tenths  ?     How  many  fifths  ? 

1.    ^  less  ^  are  how  many  ninths  ? 

227.    m.  From  f  take  -|. 

WRITTEN  rm  1  T  .  o  t        ^  P  ' 

WORK.  That  the  subtraction  may  be  performed,  these  fractionfi 

4  =  7^2"         must  be  changed  to  equivalent  fractions  having  a  com- 
■|  =  -^^         mon  denominator.     The  least  common  denominator  is  12. 

«•  \-V  i-i?  i-J?  l-i?  i-i? 
o-  5-i?  f-f?  l-f?  f-l?  «-F 


90 


COMMON  FRACTIONS. 


228.    Examples  for  the  Slate. 
In  the  following  examples,  write  the  work  as  in  122,  Art.  227. 
(45.)    t- A- what?  (49.)    17 -2J- what? 

(46.)    t-f  =  what?  (50.)    1/^-1  =  what? 

(^^')    A-TV  =  what?  (51.)    Ij-J  =  what? 

(48.)    ^T-f-what?  (52.)    121 -4f  =  what? 

229.     Examples  in  Addition  and  Subtraction. 

53.  What  are  the  contents  of  three  remnants  of  carpeting 
containing  as  follows  :  24^  yards,  S^  yards,  and  18 J  yards  ? 

54.  Clifford  had  2J  miles  to  row.  After  rowing  If  miles, 
how  much  farther  had  he  to  row  ? 

55.  A  man  sold  4J  bushels  of  cranberries  at  one  time,  3| 
bushels  at  another,  and  1^2  ^^  another.  How  many  bushels 
did  he  sell  in  all  ? 

b%.  If  it  took  12f  hours  to  plough  a  field,  and  60j  hours  to 
plant  it,  how  many  more  hours  did  it  take  to  plant  than  to 
plough  it  ? 

57.  A  farm  of  23  acres  contained  a  peach-orchard  of  4J 
acres,  a  wheat-field  of  5^  acres,  a  corn-field  of  8^  acres ;  |  of 
an  acre  was  planted  with  peas,  Ij  acres  with  strawberries,  and 
the  rest  was  a  garden.     How  much  was  a  garden  ? 

^S.  How  many  more  acres  of  the  above  farm  were  there  in 
the  corn-field  than  in  the  wheat-field  ? 

59.  In  making  some  boxes  a  carpenter  used  4J  pounds  of 
board-nails,  1^  pounds  of  shingle-nails,  and  6f  pounds  of  lath- 
nails.     How  many  pounds  did  he  use  in  all  ? 

60.  From  a  chest  of  tea  containing  42f  pounds,  17^^^  pounds 
were  sold.     How  many  pounds  were  left  ? 

61.  A  boy  once  paid  his  fare  of  $  2^^^  from  Springfield  to 
Chatham,  but,  falling  asleep,  was  carried  on  to  Albany,  where 
he  paid  %  Ij  for  his  lodging  and  breakfast,  and  $  J  g-  ^^  return 
to  Chatham.     What  was  the  entire  expense  of  his  trip  ? 

For  other  examples  in  Addition  and  Subtraction,  see  page  129. 


MULTIPLICATION  OF  FRACTIONS,  91 

MULTIPLICATION. 

To  multiply  a  Fraction  by  an  Integer. 

230.    If  James  earns  f  of  a  dollar  in  1  day,  how  much 
will  he  earn  in  2  days  ? 

Solution.  —  If  he  earns  |  of  a  dollar  in  1  day,  in  2  days  he  will  earn 

2  times  f  of  a  dollar,  which  equals  |,  or  \^  dollars.    Ans.  l\  dollars. 

In  multiplying  f  hy  2,  which  term  of  the  fraction  is  multi- 
plied? 

231.     Oral  Examples. 

a.  If  it  takes  Dana  J  of  an  hour  to  walk  a  mile^  how  long 
will  it  take  him  to  walk  2  miles  ?  3  miles  ?  6  miles  ?  8  miles  ? 

b.  How  many  dollars  will  it  take  to  give  ^  of  a  dollar  to 
each  of  4  boys  ?    5  boys  ?    8  boys  ?   9  boys  ? 

c.  How    many    are   4  times  \?    1  times  J  ?    7  times  \  ? 
7  times  f  ?    7  times  |  ?   5  times  f  ?   4  times  f  ? 

cZ.   How  many  are  6  times  |  ?    8  times  f  ?   9  times  If  ? 
Note.    To  multiply  1|,  multiply  the  integer  and  the  fraction  separately. 
e.   Multiply  4f  by  6;   7f  by  8 ;   8fby9;   9|  by  8. 

232.    Examples  for  the  Slate. 
/.   Multiply  f  by  25. 

WRITTEN    WORK 

In  doing  the  following  examples,  write  the 

3  X  25      75  o  r      > 

=  —  =  15.  work  as  it  is  written  in  Example  /.    If  there 

are  common  factors  in  the  numbers  written 

„     ^„  above  and  below  the  line,  strike  them  out  as  is 

or      3  X  2^     ^  t;  ' 

=  lo.  done  m  the  second  form  of  the  written  work. 

233.    Striking  out  equal  factors  in  the  dividend  and 
divisor  is  cancelling. 

62.  Multiply  f  by  16.  64.    Multiply  ^7_  by  15. 

63.  Multiply  I  by  63.  65.   Multiply  {}^  by  14. 


92  COMMON  FRACTIONS. 

234.  g.    Multiply  16J  by  11. 

WRITTEN 

"ifii  Multiplying  -^  by  11,  we  have  J^,  which  equals  5-|. 

2  Multiplying  16  by   11,  we  have   176,  which,   with  5^, 

gives  181^.  Ans,  ]8U. 

1I6_         66.   Multiply  5J  by  7.  68.  Multiply  161  by  10. 

181J         67.   Multiply  12§  by  12.       69.  Multiply  Q>^  by  25. 

70.  Simon  can  walk  a  mile  in  14  J  minutes.     How  long  will 
it  take  him  to  walk  3  miles  ? 

71.  If  a  lot  of  cloth  can  be  woven  in  12f\  days  on  4  looms, 
in  how  many  days  can  it  be  woven  on  1  loom  ? 

72.  Eight  persons  bought  a  tent  together,  each  paying  as 
his  share  $  15f .    What  was  the  price  of  the  tent  ? 

To  multiply  an  Integer  by  a  Fraction. 

235.  What  is  J  of  2  yards  ? 

fH — I — I — I — \ — I — I — I        If  -g-  of  each  of  2  yards  be  taken,  w^e  shall 
Teighth.  have  f  of  a  yard.  Ans.  |  of  a  yard. 

I— fH — h- H — H — I        (See  illustration  ;  also  page  58,  Example  o.) 

1  eighth. 

iof2  =  f. 

236.    Oral  Examples. 

a.  Whatisiof2?  ^ofS?  ^of3?  i^oi%%?  ^  of  8  days  ? 

b.  If  1  yard  of  tape  is  worth  3  cents,  what  is  f  of  a  yard 
worth  ? 

Solution.  —  If  1  yard  is  worth  3  cents,  f  of  a  yard  is  worth  f  of  3  cents. 
■|  of  3  is  f ,  and  -|  of  3  is  2  times  |,  which  is  |,  or  \\.       Ans.  l\  cents. 

c.  What  is  ^  of  4  ?    |  of  4  ?    f  of  14  ?    f  of  17  ?    f  of  19  ? 

d.  What  is  i  of  7  ?    |  of  7  ?    J^  of  13  ?    f  of  13  ?    |  of  17  ? 

237.    Finding  the  fractional  part  of  a  number  is  called 
multiplying  by  a  fraction. 

Thus,  in  finding  f  of  17  we  are  said  to  multiply  17  by  f . 


MULTIPLICATION  OF  FRACTIONS.  93 

238.    Examples  for  the  Slate. 

e.  Multiply  17  by  J.     /.  Multiply  24  by  /^. 

WRITTEN  WORK.  To  multiply  17  by  I  is  to  take  |  of  17. 

17  X  3  _  51  _  ^  /x ,  To  find  -|  of  17  we  may  either  divide 

^   '^     5  5  ~      ^*      ^^^'     by  5  and  then  multiply  the  result  by  3, 

4  or  we  may  first  multiply  by  3  and  then 

.^^x_5_20^g2  divide  the  result  by  5.  The  latter  method 

J^        3  is  generally  easier,  as   in  Examples   e 

^  and  /. 

73.  What  is  I  of  82?  76.    Multiply  18  by  t^^. 

74.  What  is  f  of  324  ?  77.    Multiply  30  by  ^. 

75.  What  is  T^  of  581  ?  78.   Multiply  35  by  A. 

79.  If  a  barrel  of  flour  costs  $  8.75,  what  will  f  of  a  barrel 
cost? 

80.  If  a  wheel  turns  480  times  in  going  a  mile,  how  many 
times  will  it  turn  in  going  f  of  a  mile  ? 

81.  If  a  clerk  can  copy  50  pages  in  a  day,  how  many  pages 
can  he  copy  in  2^  days  ?   in  3  J  days  ? 

82.  What  cost  14^  yards  of  cambric  at  42  cents  a  yard  ? 

83.  What  cost  16|  yards  of  silk  at  $  2.75  a  yard  ? 

To  multiply  a  Fraction  by  a  Fraction. 

239.    What  is  J  of  i  ? 
\  \  \  To  find  \  of  ^,  the  -J-  must  be  divided  into 

,      ,      I I .      2  equal  parts. 

.  Since  the  entire  unit  will  contain  3  times 

^       1  of  4  =  1.  2  or  6  such  parts,  one  of  the  parts  will  be  \ 

of  the  unit.  Ans.  \. 

In  finding  J  of  J,  how  was  the  new  denominator  obtained  ? 

240.     Oral  Examples. 

a.    What  is  i  of  i  ?   i  of  i  ?   i  of  J  ?   i  of  ^  ?  ^  of  i  ? 
5.    What  is  i  of  J  ?   i  of  J  ?   J  of  J  ?   i  of  ^  ?  i  of  J  ? 


94  COMMON  FRACTIONS. 

241.    c.   What  is  f  oft? 
WRITTEN  WORK.  J  of  |  is  ^,  (-^) ;   then  ^  of  I  must  he 

^  X  f  =  A-  A'  (5^)'  ^^^^  I  o*  f  ^^^st  ^^  2  times  as  much, 

In  finding  §  of  f ,  how  was  the  new  numerator  ohtained  ? 
the  new  denominator  ? 

To  multiply  a  fraction  by  a  fraction,  Find  the  product 
of  the  nimierators  for  the  new  mLmerator,  and  the  product 
of  the  denominators  for  the  new  denominator. 

d.  Find  §  of  i;    ioff;    ^off;    fofj-;    f  of  |. 

e,  Find  i  oft;   f  of§;    f  of  f  ;    |of§;    f  of  |. 

/.  If  a  pound  of  butter  costs  $  f^,  what  part  of  a  dollar  will 
^  of  a  pound  cost  ?     What  will  J  of  a  pound  cost  ? 

g.  A  paper-hanger  had  |  of  a  barrel  of  flour,  and  used  f  of  it 
for  paste.     What  part  of  the  barrel  did  he  use  ? 

242.    Examples  for  the  Slate. 
84.    What  is  |  of  ^%  ?  86.   What  is  |  of  ^\  ? 

S5.   What  is  |  of  /_  ?  87.    What  is  |  of  ^  ? 

In  the  following  examples  change  all  mixed  numbers  to  im- 
proper fractions  before  multiplying. 

88.  Find  f  of  f.  90.    Find  f  of  18^.         (92.)  3}  x  5J  ? 

89.  Find  f  of  14f .      91.    Find  ^V  of  3J.         (93.)  6^  x  4§  ? 

94.  What  cost  3^  barrels  of  peas  at  $  6|  a  barrel  ? 

95.  Dora  is  15f  years  old,  and  Oscar  is  f  as  old  as  Dora. 
How  old  is  Oscar  ? 

243.    Eepeat  the  following  table  : 

liong:  Measure. 

12  inches  =  1  foot.  5i  yards  or  I64  feet  =  l  rod. 

3  feet      =1  yard.  320  rods  or  5280  feet  =  l  mile. 

96.  How  many  inches  are  there  in  16^  feet  ? 

97.  How  many  feet  are  there  in  33J  yards  ? 

98.  Change  160  rods  to  yards.      99.   Change  3J  rods  to  feet. 
For  other  examples  in  Multiplication  of  Fractions,  see  page  129. 


DIVISION  OF  FRACTIONS.  95 

DIVISION. 
To  divide  a  Fraction  by  an  Integer. 

244.  If  J  of  a  melon  be  di- 
vided equally  between  2  boys, 
what  part  of  the  melon  will 
each  boy  have  ? 

If  1^  of  a  melon  be  divided  equally 
between   2  boys,    each    boy   will 
have  1  half  of  -f,  or  f  of  a  melon. 
1  half  of  I  =  |.  Ans.  f  of  a  melon. 

In  dividing  J  by  2,  what  was  done  to  the  numerator  ? 

245.    If  f  of  a  melon  be  divided  equally  between  two  boys, 

what   part   of   the   melon  wiK 
each  boy  have  ? 

As  the  number  of  parts,  3,  can- 
not be  divided  by  2  without  a  re- 
mainder, each  one  of  the  parts, 
fifths,  may  be  divided  into  two 
equal  parts  ;  these  parts  will  be 


tV  tenths  of  the  whole  melon.    There 

llialfof|=yV  will   then   be  6  tenths,  of  which 

each  boy's  part  will  be  3  tenths. 
In  dividing  f  by  2,  what  was  done  to  the  denominator  ? 

246.    To  divide  any  fraction  by  an  integer,  Divide  the 
imnierator  or  multi;ply  the  denominator  hy  the  integer. 

247.    Oral  Examples. 

a.  Divide|by2;   f  by  2  ;  f  by  3  ;  |by4;  \^hj  ^, 

b.  Divide§by3;   f  by  2  ;   f  by3;    ^by4;   t^^  by  5. 

c.  If  2  girls  divide  f  of  a  yard  of  ribbon  equally  between 
themselves,  what  part  of  a  yard  will  each  have  ? 

d.  If  9  shrubs  cost  $  2^,  what  is  the  cost  of  each  ? 

1 
^  First  change  the  mixed  number  to  an 

\   ')      t-f-      4x^""4*     improper  fraction. 


96  COMMON  FRACTIONS. 

e.  If  $  5  was  paid  for  2^  feet  of  wood,  what  was  the  cost  of 
one  foot  ? 

/.  In  a  boarding-house  7  breakfast  tickets  were  sold  for 
$  1  J,  7  dinner  tickets  for  $  1|,  and  7  supper  tickets  for  $  1  J. 
What  was  the  cost  apiece  of  each  kind  ? 

g.  A  family  put  up  8^-  bushels  of  potatoes  for  the  3  winter 
months.     What  was  that  a  month  ? 

/p.  \    3\  31         i  of  8  is  2,  with  a  remainder  of  2. 

--         2  units  equal  f ,  which,  with  ^,  are  J.     ^  of  J  is  J. 
^  ^Tis.  2|-  bushels. 

iz.  If  the  railroad  fares  of  5  persons  are  $  31  J,  what  is  the 
fare  for  each  person  ? 

i.  What  must  be  the  width  of  matting  of  which  it  takes  5 
breadths  to  reach  8|  yards  ? 

j.  What  must  be  the  width  of  paper  of  wliich  it  requires 
10  breadths  to  reach  24 J  feet  ? 

k.   Divide  If  by  6;    2f  by  4 ;   4i  by  3 ;   8|  by  11. 

L    Divide  ISJ  by  4 ;   by  5 ;    by  6  ;   by  7 ;   by  8  ;   by  9. 

248.    Examples  for  the  Slate, 

100.  Mrs.  Grant  gave  $  10 J  for  27  Christmas  wreaths. 
How  much  did  she  give  apiece  ? 

101.  A  gentleman  paid  $  67 J^  for  board  for  120  days.  What 
was  the  price  per  day  ? 

102.  If  1  needle-woman  can  do  a  piece  of  work  in  11^  days, 
in  what  time  can  12  do  it  ?     In  what  time  can  10  do  it  ? 

103.  If  a  certain  field  can  be  reaped  in  15f  days,  in  what 
time  could  a  field  that  is  -^^  as  large  be  reaped  ? 

-  104.    If  119J  yards  of  cambric  are  required  to  make  9  dresses, 
how  much  is  required  for  1  dress  ? 

105.  If  a  railroad  car  runs  122/ij  miles  in  5  hours,  what  is 
the  average  rate  an  hour  ? 

106.  Divide  ^Vt  by  25.  (109.)   363.-11. 

107.  Divide  II  by  26.  (110.)    16f-f-20. 

108.  Divide  tl  by  110.  (111.)   2724-18. 


DIVISION  OF  FRACTIONS.  97 

To  divide  an  Integer  or  a  Fraction  by  a  Fraction. 
249.     Oral  Examples. 

a.  In  one  there  are  how  many  halves  ?    thirds  ?   fourths  ? 

b.  In  two  there  are  how  many  halves  ?    thirds  ?   fourths  ? 

c.  Julian  had  5  cents,  which  he  spent  for  pears  at  J  cent 
apiece.     How  many  pears  did  he  buy  ? 

d.  How  many  pears  could  he  buy  at  J  of  a  cent  apiece  ?  at 
J  of  a  cent  apiece  ? 

e.  How  many  mugs  at  $  ^  each  can  you  buy  for  $  3  ?  for  $  5  ? 

250.  /.  Helen  made  4  knots  of  worsted  into  tassels,  putting 
§  of  a  knot  into  each  tassel.    How  many  tassels  did  she  make  ? 

She  must  have  made  as  many  tassels  as 

WRITTEN   WORK.  ^,  ^.  o   -       a 

there  are  times  f  m  4. 

(/.)  4  ==  \^-  We  first  change  4  to  thirds,  making  J^. 

1  2.  ^  2  _  ][2  -f-  2  =  6.       There  are  as  many  times  f  in  ^  as  there 

are  2's  in  12,  or  6.  ^ns.  6  tassels. 

Note.    For  a  different  analysis  of  this  example,  see  Appendix,  page  141. 

g.    Miss  Breck  made  6  yards  of  crash  into  towels,  putting  J 
of  a  yard  in  each  towel.     How  many  did  she  make  ? 
h.    Divide  6  by  f;    by  f  ;    by  ,^  ;    by  |  •    by  #. 
i.    Divide  8  by  ^  ;  by  f ;    by  1^  or  | ;    by  2^  or  J. 
J.    How  many  hats  at  $  1 J  each  can  be  bought  for  $  12  ? 

How  do  you  change  an  integer  to  divide  it  by  a  fraction  ? 
How  do  you  then  divide  ?  How  do  you  divide  by  a  mixed 
number  ? 

251.  k.   Divide  f  by  f .         o.   2f  are  how  many  times  f  ? 
i.    Divide  |  by  f .  i?.    4f  are  how  many  times  f  ? 
122.  Divide  f  by  f .  g.    3|  are  how  many  times  1\? 
n.   Divide  3  J  by  g.                  -^-    ^i  a-re  how  many  times  2 J  ? 
s*.    In  a  farm-school  5\  acres  of  land  were  given  to  some  boys 

to  plant,  each  boy  having  \  of  an  acre.    To  how  many  boys  was 
the  land  given  ? 

When  fractions  have  a  common  denominator,  how  do  you 
divide  ? 


98  COMMON  FRACTIONS, 

252,  t.  How  ma,ny  feet  of  rope  can  be  made  from  1^  pounds 
of  hemp,  there  being  J  of  a  pound  to  a  foot  ? 

WRITTEN  wo  UK.  As  many  feet  can  be  made  as  there 

(t.)  13  =  I  =  fi  ;   i  -  tV.         are  times  J-  in  If  or  |. 

2 1  _^  2  =  21  -  2  =  101-         i  ^^^  6  changed  to  fractions  liaving 
^  ^*     a  common  denominator  are  2^  and  ^2^. 

f^  divided  by  ^  equals  21  divided  by  2,  or  10^.     Ans,  10^  feet. 

When  fractions  have  different  denominators,  how  do  you 
prepare  them  to  divide  ?     How  do  you  then  divide  ? 

253.  To  divide  an  integer  or  a  fraction  by  a  fraction, 
Change  the  dividend  and  divisor  to  fractions  having  a 
common  denominator,  and  then  divide  the  numerator  of  the 
dividend  hy  the  numerator  of  the  divisor.  (See  Appendix 
page  141.) 

12.    How  many  times  is  f  contained  in  f  ?    f  in  §  ?    |  in  f  ? 

V,    How  many  times  is  -^^  contained  in  f  ?  f  in  J  ?  J  in  1 J  ? 

w.  I  have  a  dish  which  holds  §  of  a  quart.  How  many  times 
must  it  be  filled  to  measure  2^  quarts  ? 

X.  How  many  house-lots  each  containing  \  of  an  acre  can 
be  made  from  a  lot  containing  2  J  acres  ? 

254.    Examples  for  the  Slate. 

112.  What  is  36-^^?  117.    Divide  ^  by  t^. 

113.  What  is  48 -IJ?  118.    Divide  f  by  IJ. 
114    What  is  52 -3i?  119.    Divide  33 i  by  4 J. 

115.  What  is  100-61?  120.    Divide  If  by  7f. 

116.  What  is  4^ -1|?  121.    Divide  8 J  by  4J-. 

122.  How  many  lengths  of  fencing  f  of  a  rod  long  will  it 
take  to  enclose  a  square  lot  of  land,  each  side  40  rods  long  ? 

123.  The  rations  of  a  ship's  company  were  Ij  pounds  of 
meat  a  day  to  each  person,  but  after  losing  a  part  of  their  pro- 
visions in  a  storm,  they  were  allowed  only  -f^  as  much.  What 
were  their  rations  then  ? 

For  other  examples  in  Division,  see  page  129. 


COMMON  FRACTIONS.  99 

To  find  the  "Whole  when  a  Part  is  given. 
OrcQ  Eszercises. 

255.  a.  If  J  pound  of  yarn  costs  12  cents,  what  will  a 
pound  cost  at  the  same  rate  ? 

b.  In  \  of  a  ream  of  paper  there  are  5  quires.  How  many- 
quires  are  there  in  a  ream  ? 

c.  Mr.  Fitch  sold  a  boat  and  gained  $  8,  which  was  \  of 
what  it  cost  him.     How  much  did  it  cost  him  ? 

d.  8  is  \  of  what  number  ? 

e.  A  man  sold  a  wagon  and  lost  $  11,  which  was  ^  of  what 
it  cost  him.     What  did  it  cost  him  ? 

/.    If  ^  of  a  yard  of  silk  cost  $  1^,  what  will  1  yard  cost  ? 

g.  From  a  cask  of  oil  15  gallons  leaked  out.  If  this  was 
-fV  of  the  contents  of  the  cask,  what  did  the  cask  contain  at 
first? 

h.  Frank  has  a  sister  whose  age  equals  f  of  his  own  age. 
If  the  difference  between  their  ages  is  2  years,  what  is  Frank's 
age  ? 

Suggestion.  Frank's  age  is  -|  of  itself.  The  difference  between  -f 
and  f ,  which  is  \,  is  2  years» 

i.  After  losing  J  of  his  money,  a  man  had  $  12  left.  How 
much  money  had  he  at  first  ? 

j.  A  man  traded  off  a  cow  and  got  |  of  what  she  cost  him. 
If  he  gained  $  5,  what  did  she  cost  him  ? 

k.  The  difference  between  ^  and  ^  of  a  bushel  is  8  quarts. 
How  many  quarts  are  there  in  a  whole  bushel  ? 

L  The  difference  between  f  and  y%  of  a  number  is  5.  What 
is  that  number  ? 

m.  The  difference  between  J  and  ^  of  a  number  is  Ij.  What 
is  that  number  ? 

n.    $  5  is  .^  of  how  many  dollars  ? 

0.    1 J  is  J  of  what  number  ?    \  of  what  number  ? 

p.    65  is  J  of  what  number  ?    J  of  what  number  ? 


100  COMMON  FRACTIONS. 

256.  a.  If  f  of  a  piece  of  work  can  be  performed  in  4  days, 
in  how  many  days  can  ^  of  the  work  he  performed  ?  In  how 
many  days  can  the  whole  work  be  performed  ? 

b.  If  f  of  a  number  is  4,  what  is  the  whole  number  ? 

Solution.  —  If  f  of  a  number  is  4,  \  of  the  number  is  \  of  4,  which 
is  2,  and  \  of  the  number  is  5  times  2,  or  10.  Ans.  10. 

c.  A  man  performed  J  of  a  journey  in  15  days.  In  how 
many  days  could  he  perform  \  of  the  journey  ?  In  how  many 
days  could  he  perform  the  whole  journey  ? 

d.  If  f  of  a  number  is  15,  what  is  the  whole  number  ? 

e.  If  f  of  a  quire  of  paper  cost  10  cents,  what  did  the  whole 
quire  cost  ? 

/.    f  of  a  certain  number  is  10.     What  is  that  number  ? 

g.  If  there  are  21  lines  in  ^  of  a  page,  how  many  lines  are 
there  in  the  whole  page  ? 

h,   21  is  J  of  what  number  ? 

i.  A  grocer  sold  eggs  at  20 /'  a  dozen,  which  was  |  of  what 
they  cost  him.     What  did  they  cost  him  ? 

J.  A  man  gained  $  15  by  selling  candy  for  J  as  much  money 
as  it  cost  him.     What  did  it  cost  ? 

k.    15  is  I  of  what  number  ? 

i.  Edgar  sold  a  kite  for  f  of  what  it  cost  him,  and  thereby 
lost  8  cents.     What  did  the  kite  cost  him  ? 

222.  8  is  f  of  what  number  ? 

22.  By  selling  beef  at  6J  ^  a  pound,  a  market-man  gets  but  g 
of  what  it  cost  him.     How  much  did  it  cost  him  ? 

o.    6J  is  f  of  what  number  ? 

p.  After  I  of  a  certain  month  is  past  there  are  24  days  re- 
maining.    How  many  days  are  there  in  the  month  ? 

q.  A  farmer  finds  after  selling  f  of  his  pears  that  he  has  8 J 
barrrels  left.     How  many  barrels  had  he  at  first  ? 

r.    8J  is  f  of  what  number  ?  iz.    J  is  f  of  what  ? 

s.  1-^y  is  f  of  what  number  ?  t^.   ^  is  f  of  what  ? 

t.  4J  is  j%  of  what  number  ?  w.  ^  is  f  of  what  ? 


MISCELLANEOUS  EXAMPLES.  101 

257.    Miscellaneous  Oral  Examples. 

a.  At  $  J  a  day  for  board,  how  many  daj^s'  board  may  be 
obtained  for  $  7 J  ? 

b.  If  }  of  a  cord  of  wood  costs  $  6,  what  will  1  cord  cost  ? 

c.  If  5  men  can  do  a  piece  of  work  in  f  of  a  week,  in  what 
time  can  1  man  do  it  ? 

d.  A  man  bought  a  quantity  of  drugs  for  $53,  and  was 
obliged  to  sell  them  for  -f^  of  what  they  cost  him.  How  much 
did  he  receive  for  them  ? 

e.  A  miller  paid  90  cents  a  bushel  for  wheat,  and  sold 
it  so  as  to  lose  §  as  much  as  it  cost.  What  did  he  receive 
for  it  ? 

/.  If  f  of  a  bale  of  cotton  cost  %  30,  what  will  1  bale  cost  ? 
How  many  yards  of  cloth  at  $  5  a  yard  will  pay  for  a  bale  of 
cotton  ? 

g.    If  9  bottles  of  ink  cost  $  2\,  what  will  5  bottles  cost  ? 

h.  If  6  pounds  of  lead  cost  $  1  J,  what  will  7  pounds  cost  ? 

i.  How  many  packages  of  f  of  a  pound  each  can  be  made 
from  9  pounds  of  starch  ? 

j.    If  ^  of  a  barrel  of  oil  cost  $  36,  what  cost  J  of  a  barrel  ? 

k.   If  $  I  will  buy  6  gallons  of  oil,  how  much  will  $  |  buy  ? 

1.  If  20  cents  are  paid  for  f  of  a  basket  of  peaches,  what  is 
the  cost  of  f  of  the  basket  ? 

122.  If  §  of  a  box  of  raisins  is  worth  $  4,  how  many  clocks  at 
$  2  apiece  will  pay  for  a  whole  box  ? 

22.  Mr.  Wing  sold  6  pigs  for  $  24,  which  was  f  of  what  he 
paid  for  them.     What  did  he  pay  apiece  for  them  ? 

0.  If  a  man  by  travelling  8  hours  a  day  can  perform  a  jour- 
ney in  If  days,  in  how  many  days  can  he  perform  it  by  travel- 
ling 10  hours  a  day  ? 

p.  I  sold  a  horse  for  $  48,  which  was  f  of  what  he  cost  me. 
When  I  bought  the  horse  I  paid  for  him  with  hay  at  $  10  a 
ton.     How  many  tons  did  it  take  ? 

q.  If  a  man  can  walk  at  the  rate  of  |  of  a  mile  in  7J  min- 
utes, in  What  time  can  he  walk  1  mile  ?    5  miles  ? 


102  MISCELLANEOUS  EXAMPLES. 

258.    Miscellaneous  Examples  for  the  Slate. 

124.  Mr.  Sage  lias  4  pieces  of  velvet ;  the  first  contains  7| 
yards,  the  second  8^  yards,  the  third  5/(j  yards,  and  the  fourth 
8]^  yards.     How  many  yards  are  there  in  all  ? 

125.  At  $  5  J  a  ream,  what  cost  ^  of  a  ream  of  paper  ? 

126.  Mr.  Gray  bought  18^  acres  of  land,  and  gave  1  half  of 
it  to  his  son  and  1  lialf  of  the  remainder  to  his  nephew.  How 
much  did  he  give  to  his  nephew  ? 

127.  Emma's  dress  cost  $11|,  and  Mary's  g  as  much.  What 
did  Mary's  cost  ? 

128.  Mr.  Brown  had  a  huilding-lot  that  measured  12^  rods 
m  front.     What  did  it  measure  in  yards  ?    in  feet  ? 

129.  How  many  rods  of  fencing  on  both  sides  of  a  road  that 
is  1  J-  miles  long  ? 

130.  What  is  the  cost  of  fencing  a  road  which  is  2  of  a  mile 
in  length,  at  $  14i-  a  rod  ? 

131.  Olive  had  111'^.  If  she  spent  all  but  $5^^  of  it  for 
cambric  at  $  |^  a  yard,  how  many  yards  did  she  buy  ? 

132.  Winthrop  received  $  18.10  for  work,  and  after  j)aying 
a  debt  of  $  4.80,  had  money  enough  to  pay  for  3 J  weeks'  board. 
What  was  the  price  of  board  per  week  ? 

133.  Mr.  Channing  bought  land  for  $  1  If  an  acre,  and  sold 
it  for  $  19 J  an  acre.    What  did  he  gain  on  1  acre  ?  on  3  acres  ? 

134.  Mr.  Howes  sold  12  dozen  cabbages  at  8J/  apiece,  and 
received  in  payment  46  pounds  of  beef  at  9  >^  a  pound,  and  the 
rest  in  money.     What  did  he  receive  in  money  ? 

135.  How  many  dominos  can  be  cut  from  a  piece  of  ivory 
7J  inches  long,  allowing  Ij  inches  for  each  domino  ? 

136.  How  many  checkers  can  be  made  from  a  block  15  inches 
long,  if  f  of  an  inch  is  allowed  for  each  checker  ? 

137.  A  man  owned  23f  acres  of  meadow,  11\  acres  of  pas- 
ture, and  5J  acres  of  woodland.  How  many  acres  did  he  own 
in  all  ? 

138.  What  will  25^^  yards  of  carpeting  cost  at  $1|  per 
yard  ? 


MISCELLANEOUS  EXAMPLES.  103 

139.  At  $4J  per  week,  wliat  is  the  cost  of  board  for  3^ 
weeks  ? 

140.  How  man}'-  rations  of  Ij  pounds  each  can  be  furnished 
from  a  barrel  of  beef  weigliing  200  pounds  ? 

141.  Three  boys  started  together  to  skate  down  a  river. 
One  skated  a  mile  in  8 J-  minutes,  another  in  10  J  minutes,  and 
the  third  in  9J-  minutes.  How  much  longer  did  it  take  the 
second  than  the  first  ?    than  the  third  ? 

142.  Tliese  boys  started  together  to  return,  and  found  that 
it  took  them  11 J  minutes,  12  §  minutes,  and  13 J^  minutes  re- 
spectively, to  skate  the  mile.  How  many  minutes  had  the  first 
to  rest  before  the  second  came  up  ?  How  many  before  the 
third  came  up  ? 

143.  When  1  pound  of  cotton  is  put  into  3|  yards  of  cloth, 
how  much  is  put  into  half  a  yard  ?   into  12^  yards  ? 

144.  If  30  yards  of  cloth  1  yard  wide  weigh  12%  pounds, 
what  will  30  yards  of  the  same  kind  of  cloth  weigh  if  it  is  J  of 
a  yard  wide  ?    if  it  is  |  of  a  yard  wide  ? 

145.  After  going  J  of  his  journey,  Mr.  Otis  had  32|  miles  to 
go.     What  was  the  length  of  his  journey  ? 

146.  Homer's  saddle  cost  $25,  which  was  -^^  of  what  his 
horse  cost.     What  did  his  horse  cost  ? 

147.  A  man  sold  a  watch  for  $  18^,  which  was  f  of  what 
he  gave  for  it.     What  did  he  give  for  it  ? 

148.  Mr.  Bracket  sold  a  coat  and  gained  $  1.32,  which  was 
\  of  what  it  cost  him.     What  did  it  cost  him  ? 

149.  If  5  yards  of  cloth  cost  $  2.25,  what  will  7  J  yards  cost  ? 

150.  If  14  cedar  posts  cost  $8|,  what  will  10  cost  ? 

151.  If  ^  pounds  of  lead  cost  $  1.70,  what  will  5J  pounds 
cost? 

152.  A  man  sold  a  iirkin  of  butter  for  $  16.80,  and  gained  J 
of  what  it  cost  him.     What  did  it  cost  him? 

153.  On  counting  his  money,  Mr.  Gould  found  he  had 
$  36.72.  If  this  was  f  as  much  as  he  had  spent,  how  much  had 
he  spent  ?     How  much  had  he  at  first  ? 


104 


DECIMAL  FRACTIONS. 


SEOTION^X, 


< 


< 

a 


DECIMAL    FRACTIONS. 

259.  a.    If  a  block  be  divided  into  ten  equal  parts,  what 

is  each  of  the  parts 
called  ? 

b.  If  one  of  these 
tenths  be  divided 
into  ten  equal 
parts,  what  will 
each  of  these  parts 
be  called  ?      What 

is^Joof  1^0? 

c.  If  one  of  these  hundredths  be  divided  into  ten  equal  parts, 
what  will  each  of  these  parts  be  called  ?     What  is  -^  of  -^^-q  ? 

d.  How  many  tenths  make  a  unit  ? 

e.  How  many  hundredths  make  a  tenth  ? 

/.     How  many  thousandths  make  a  hundredth  ? 

g.    What  part  of  a  tenth  is  a  hundredth  ? 

h.    What  part  of  a  hundredth  is  a  thousandth  ? 

One  tenth  of  a  thousandth  is  a  ten-thousandth;  one 
tenth  of  a  ten-thousandth  is  a  hundred-thousandth ;  one 
tenth  of  a  hundred-thousandth  is  a  millionth;  and  so  on. 

260.  The  fractions,  tenths,  hundredtJis,  thousandths ,  etc., 
are  decimal  fractions,  or  decimals. 


To  read  and  write  Decimal  Fractions. 

261.  Decimal  fractions  may  be  written  like  common 
fractions,  with  denominators.  But  they  are  usually  written 
in  the  same  Avay  that  integers  are  written.    To  distinguish 


DECIMAL  FRACTIONS. 


105 


them  from  integers,  a  point  (.),  called  a  decimal  point,  is 
put  at  the  left  of  the  tenths'  place. 
Thus  we  write 


One  tenth 0.1 

One  hundredth 0.01 

One  thousandth 0.001 


Three  tenths 0.3 

Two  hundredths 0.02 

Five  thousandths 0.005 


Three  tenths  and  two  hundredths,  or  thirty-two  hundredths....  0.32 
Two  hundredths  and  five  thousandths,  or  twenty-five  thousandths  0.025 

Three  tenths,  two  hun- 
dredths, and  five 
thousandths,  or  three 
hundred  twenty-five 
thousandths 0.325 

One  and  three  hundred 
twenty-five  thou- 
sandths   1.325 


It  will  be  seen 
that  in  this  method  of  writing  decimals  it  is  the  numerator 
only  that  is  written.  How  is  the  denominator  indicated  ? 
Ans.  The  denominator  is  indicated  hy  the  place  of  the  right- 
hand  figure,  counting  from  the  decimal  point. 

262.  The  method  of  writing  decimals  is  more  fully 
shown  by  the  following  table,  which  is  merely  an  extension 
of  the  table  given  on  page  73. 


.1 


$    .9 
0     . 


I 

4 


I 

I 

9 


I 


In  which  place  from  the  right  of  the  decimal  point  are  ten- 
thousandths  written  ?    hundred-thousandths  ?    millionths  ? 


106  DECIMAL   FRACTIONS. 

263.     Exercises. 


Eead  the  following  : 

a.        b. 
0.4       0.04 

6. 

d. 

0.001 

e. 
41.5 

0.9 

0.36 

0.5 

0.668 

0.06 

4.6 

8.88 

3.21 

3.704 

60.209 

3.2 

7.59 

6.13 

20.032 

0.006 

7.7 

3.40 

25.07 

6.006 

13.013 

0.0001 

6.2724 

h. 
68.5 

i. 
1324. 

0.0045 

0.3662 

0.007 

132.4 

0.0182 

0.3079 

2.904 

13.24 

0.3247 

17.3761 

0.8' 

764 

1.324 

26.0026 

42.6305 

279.0017 

0.1324 

Turn  to  page  129,  and  read  the  numbers  written  in  H  and  I. 

264.    Examples  for  the  Slate. 

Write  in  figures  the  following  numbers  : 

1.  Eight  tenths.  (6.)  4  units  and  5  hundredths. 

2.  Seven  hundredths.  (7.)  274  units  and  8  tenths. 

3.  Twelve  thousandths.         (8.)  906  hundred-thousandths. 

4.  Thirty-four  hundredths.   (9.)  4  units  and  7  ten-thousandths. 

5.  Eight  ten-thousandths.  (10.)  2148  hundred-thousandths. 
Let  the  teacher  dictate  other  decimals  for  the  pupils  to  write. 

REDUCTION  OF  DECIMALS. 
To  change  a  Decimal  to  lo"wer  Denominations. 

265.    a.    Change  3  to  tenths,  b.   0.5  to  hundredths;    to 
thousandths. 

WRITTEN  WORK.  rpo  Gxpress  a  decimal 

3  =  3.0       read  30  tenths.  fraction  in  any  lower  de- 

0.5  =  0.50     read  50  hundredths.  nomination,  Annex  zeros 

0.5  =  0.500  read  500  thousandths.  ^^    ^^^    ^.^^.^   .a:^r...^.^ 

until  the  place  of  the  required  denomination  is  reached. 


REDUCTION  OF  DECIMALS.  107 

266.    Examples  for  the  Slate. 

11.  Change  7  to  tenths  ;    to  hundredths  ;    to  thousandths. 

12.  Change  5  to  thousandths ;    0.25  to  ten-thousandths ; 

To  change  Decimals  to  Common  Fractions. 

267.   c.  Change  0.5  to  a  common  fraction  in  its  simplest  form. 

wrjTTEN  WORK.  To   changG  a  decimal  to  a  common 

(c.)   0.5  =  /w  =  o*      f Inaction,  First  ivrite  the  fraction  luith  its 

denominator;    then  change  the   common 

fraction  to  its  smallest  terms. 

268.     Examples   for  the   Slate. 

Change  the  following  to  common  fractions  in  their  simplest 
forms : 

(13.)   0.2;   0.4;   0.5;   0.6;   0.7;   0.05;   0.15;   0.25;   0.75. 
(14.)   0.125;   0.375;    0.625;   0.875. 

To  change  Common  Fractions  to  Decimal  Fractions. 
269.    d.    Change  |  to  a  decimal  fraction. 
WRITTEN  WORK.  The  fraction  |  is  the  same  as  \  of  3,  or  |  of 

(d.)    8)  3.000  3.000,  which  is  found  hy  dividing  3.000  by  8  in 

0.375  the  usual  w^ay. 

To  change  a  common  fraction  to  a  decimal,  Express  the 
nnmerator  in  tenths,  hundredths^  thousandthsy  etc.,  hy  an-, 
nexing  as  many  zeros  as  may  he  reqitired,  and  then  divide 
it  hy  the  denomincctor. 

^.  270.    Examples  for  the   Slate. 

Change 

(15.)  ^  to  tenths.  (19.)  J  to  hundredths.    (23.)  i  to  thousandths. 

(16.)  i  to  tenths.  (20.)  f  to  hundredths.    (24.)  f  to  thousandths. 

(17.)  i  to  tenths.  (21.)  ^7^  to  hundredths.  (25.)  J  to  thousandths. 

(18.)  I  to  tenths.  (22.)  ^  to  hundredths.    (26.)  ^  to  thousandths. 

For  other  examples,  see  page  129. 


108 


ADDITION  OF  DECIMALS. 


ADDITION    OF    DECIMALS. 
271.     B»^mples   for  the   Slate. 


Write  and  add  the  following ; 


WRITTEN   WORK. 

5.7 
13.8 
9.14 
6.27 


a. 

5  and    7  tenths, 
13  and    8  tenths, 

9  and  14  hundredths, 

6  and  27  hundredths. 

Ans.    34.91 

(27.)   Add 

3  and    7  tenths, 
75  and    8  tenths, 

6  and  43  hundredths, 
16  and    9  hundredths. 

(28.)    Add 
^S  and  72  hundredths, 

3  and    9  tenths, 
43  and  46  hundredths, 
7  and    7  hundredths. 
200  and    6  hundredths. 

(29.)    Add 
8  and  19  hundredths, 
b^  and    9  tenths, 

b^  hundredths, 
24  and    8  hundredths, 
9  tenths. 

(30.)    Add 

5  and  126  thousandths, 
14  and  374  thousandths, 

276  and  11  thousandths, 
489  thousandths, 

6  and  108  thousandths. 


In  adding  decimals, 
Add  as  in  integers,  fixing 
the  decimal  point  in  the 
answer  as  soon  as  the 
tenths  are  added. 


(31.)   Add 

1  and  926  thousandths, 
754, 

37  and    47  hundredths, 
8  and    78  thousandths. 

(32.)   Add 

5  and  4763  ten-thousandths, 

21  and  2131  ten-thousandths, 

3245  ten-thousandths, 

9  and    875  ten-thousandths, 

47  and      26  thousandths. 

(33.)    Add 
14  and  3748  ten-thousandths, 

4  and      93  hundredths, 
7  and    872  thousandths, 

5  and        8  tenths, 

9875  ten-thousandths. 

(34.)   Add 

7  and  1  hundredth, 

6^25  ten-thousandths, 
875  thousandths, 
46  and      97  ten-thousandths, 

8  and  1008  ten-thousandths. 


SUBTRACTION  OF  DECIMALS.  109 

35.  Add  twelve  and  three  tenths ;  seventy  and  thirty-five 
hundredths ;  two  thousand  three  hundred  forty-seven  ten- 
thousandths  ;    and  seventy-eight  thousandths. 

36.  Add  forty-seven  and  three  hundred  seventy-eight  thou- 
sandths ;  twenty-six  ten-thousandths  ;  four  and  nineteen  hun- 
dredths; and  sixty-six  thousand  six  hundred  and  sixty-six 
hundred-thousandths. 

37.  Turn  to  page  106,  and  add  the  numbers  written  in 
Exercises  a  to  i. 

For  other  examples  in  Addition,  see  page  129. 

SUBTRACTION    OF    DECIMALS. 

272.  Examples  for  the  Slate. 

a.   From  2  and  75  hundredths  take  928  thousandths. 

WRITTEN  WORK.  In  subtiacting  decimals,  Subtract  as  in 

2.75  integers,  fixing   the   decimal  point   in    the 

Q'^^^  remainder  as  soon  as  the  tenths  are  sub- 

Ans.  1.822  tr  acted. 

38.  Prom  14  and  5  tenths  take  75  hundredths. 

39.  From  7  and  43  hundredths  take  647  thousandths. 

40.  From  675  thousandths  take  497  thousandths. 

41.  Subtract  445  thousandths  from  2  and  4  tenths. 

42.  Subtract  6^  thousandths  from  6  and  6  tenths. 

43.  Subtract  9  ten-thousandths  from  9  hundredths. 
Find  the  difference  of  the  following : 

(44.)  (45.)  (46.)  (47.)  (48.)  (49.) 

4.075        75.09        10.066         0.202  27.9384  1.11 

2.439  7.446        0.0066       0.0202  3.42853        0.9999 

273.  Miscellaneous   Examples. 

50.  A  dozen  lemons  cost  %  0.38 ;  four  pounds  of  sugar, 
$  0.36 ;  and  three  pounds  of  crackers,  $  0.25.  What  did  all  cost  ? 

51.  One  eighth  of  a  dollar  is  $  0.125  ;  five  eighths  is  $  0.625. 
What  is  the  sum  expressed  decimally  ?    What  is  the  difference  ? 


110  MULTIPLICATION  OF  DECIMALS. 

52.  If  from  a  cask  containing  15.4  gallons  of  ink  7.875  was 
drawn  out,  how  much  remained  ? 

53.  If  the  ages  o^JPbhn's  grandfathers  are  71.125  and  72.5 
years,  and  of  his  grandmothers  69.33  and  70.25  years,  what  is 
the  sum  of  all  their  ages  ? 

54.  What  is  the  difference  between  the  ages  of  John's  grand- 
fathers ?   between  the  ages  of  his  grandmothers  ? 

For  other  examples  in  Subtraction,  see  page  129. 

MULTIPLICATION  OF  DECIMALS. 
To  multiply  a  Decimal  by  an  Integer. 
274.    a.   Multiply  0.3  by  4  ;   0.03  by  4;   0.003  by  4. 
WRITTEN  WORK.  Three  tenths  multiplied  by  4  is  12 

0.3       0.03       0.003  tenths ;   three  hundredths  multiplied  by 

4  4  4  4  is  12  hundredths  ;   three  thousandths 

1.2      0.12      0.012  multiplied  by  4  is  12  thousandths. 

In  multiplying  a  decimal  by  an  integer,  how  many  places 
for  decimals  do  you  point  off  in  the  product  ? 

275.    Examples  for  the  Slate. 

m.   Multiply  0.4  by  9  ;   0.04  by  9  ;   0.004  by  9. 
m.    Multiply  1.2  by  12  ;    0.12  by  12  ;   0.024  by  12. 

57.  Multiply  7.5  by  15  ;   6.25  by  5;   87.5  by  20. 

58.  What  is  the  cost  of  5  pounds  of  butter  at  $0.3  per 
pound  ?   at  $  0.4  ?   at  %  0.35  ?   at  S  0.40  ? 

59.  There  are  16.5  feet  in  one  rod.  How  many  feet  in  a 
mile,  or  320  rods  ? 

To  multiply  an  Integer  by  a  Decimal. 

276.   b.  What  is  tV  of  7?  yV^f^?  yio  of  ^  ?  ^l^oi25? 

c.   Multiply  11  by  0.1 ;   by  0.01 ;   by  0.12. 

WRITTEN  WORK.  To  multiply  11  by  0.1  is  to  take  1  tenth 

11  11  11  of  it,  which  we  express  by   placing  the 

0.1       0.01       0.12  decimal  point  so  that  the  figures  11  may 

T~l        Q 11        1  32  express  tenths,  thus,  1.1. 


MULTIPLICATION  OF  DECIMALS.  Ill 

To  multiply  11  \>y  0.01  is  to  take  1  hundredtli  of  it,  which  we 
express  by  placing  the  decimal  point  so  that  the  figures  may  express 
hundredths;  thus,  0.11. 

To  multiply  11  by  0.12  is  to  take  12  hundredths  of  it.  One  liundredth 
of  11  is  0.11,  and  12  hundredths  is  12  times  0.11,  which  equals  1.32. 

Aiu.  1.32. 

In  multiplying  an  integer  by  a  decimal,  how  many  places 
for  decimals  do  you  point  off  in  the  product  ? 

277.    Examples  for  the  Slate. 

60.  Multiply  9  by  0.2;   by  0.02 ;   by  0.05. 

61.  Multiply  12  by  0.6;   by  0.08;   by  0.008. 

62.  Multiply  25  by  0.12 ;   by  0.012;   by  1.2. 

63.  At  $8  a  ton  for  coal,  what  cost  0.9  of  a  ton?  0.09  of 
ton?   5.4  tons?   7.23  tons  ? 

To  multiply  a  Decimal  by  a  Decimal. 

278.  d.  What  is  J^  of  tV  ^  tV  of  t\  ?  ^js  of  A^  A  of  ^7^? 
e.  What  is  jV  of  jgi)  ?   0.1  of  0.05  ?   0.4  of  0.06  ? 

/.    What  is  tJo  of  tV  ^   0.02  of  0.7  ?   0.06  of  0.9  ? 
g.  Multiply  0.5  by  0.9.     h.  Multiply  0.5  by  0.09. 

WRITTEN  WORK.  {g.)  To  multiply  0.5  by  0,9  is  to  take  9  tenths 

{g.)  (h.)  of  5  tenths.     One  tenth  of  5  tenths  is  5  hun- 

0.5  0.5  dredths,  and  9  tenths  is  9  times  5  hundredths, 

0.9  0.09  which  is  45  hundredths,  0.45. 

"TTT^  0~04^  (^^•)  ^^^  ^^^  same  way  we  find  that  0.5  multi- 

plied by  9  hundredths  gives  45  thousandths. 
As  it  requires  three  decimal  places  to  write  thousandths,  and  as  in 
45  there  are  but  2,  we  supply  the  deficiency  by  prefixing  a  zero  before 
placing  the  decimal  point,  thus,  0.045. 

In  multiplying  a  decimal  by  a  decimal,  how  many  decimal 
places  do  you  point  off  in  the  product  ? 

279.  In  multiplying  by  decimals,  Multiply  as  in  in- 
tegers, and  point  off  as  many  j^^^f^es  for  decimals  in  the 
product  as  there  are  decimal  p)laces  in  the  midtiplicand  and 
multiplier  counted  together. 


112  DIVISION  OF  DECIMALS. 


^ 

280.     Examples 

for  the 

Slate. 

64. 

Multiply  11.6  by  4. 

71. 

30.04x0.105  = 

9 

^^, 

Multiply  1.16  by  7. 

72. 

920.8x706.1- 

9 

m. 

Multiply  17.07  by  11. 

73. 

3.007  X  0.005  = 

9 

67. 

Multiply  106  by  0.3. 

74. 

5005.  X  0.001  = 

9 

68. 

Multiply  20  by  1.71. 

75. 

88.04  X  36  =  ? 

69. 

Multiply  3.21  by  28. 

76. 

0.325  X  0.018  = 

9 

70. 

Multiply  30.2  by  1.4. 

77. 

0.0101  X  4.16  - 

9 

78.  At  $0.62^  ($0,625)  each  for  hats,  what  is  the  cost  of 
10  hats  ?   of  2  dozen  ? 

79.  At  $  0.20  a  square  foot  for  land,  what  is  the  cost  of  1000 
square  feet?   of  2500  ? 

80.  What  cost  124^  yards  of  cloth  at  1 1.875  a  yard  ? 

81.  What  must  be  paid  for  100  feet  of  land  at  $  0.06^  a  foot  ? 

82.  There  are  30.25  square  yards  in  a  square  rod.     How 
many  square  yards  are  there  in  160  square  rods  or  an  acre  ? 

DIVISION. 
To  divide  a  Decimal  by  an  Integer. 

281.    a.  What  is  1  third  of  0.6  ?   1  fourth  of  0.8  ?   of  1.2  ? 

b.  What  is  1  half  of  0.04  ?  1  fifth  of  0.25  ?  1  eighth  of  0.56  ? 

c.  Divide  13.75  by  5.     d.  Divide  10.272  by  4. 

WRITTEN  WORK.  Ill  Example  c,  the  dividend  has  two 

(c.)  (cf.)  places  for  decimals,  so  in  the  quotient 

5)  13.75  4)  10.272  we  mark  off  two  places  for  decimals. 

~2T5  ~~2  568  ^^  Example    d,   the    dividend  has 

three  places   for  decimals,   so   in  the 

quotient  we  mark  off  three  places  for  decimals. 

In  dividing  a  decimal  by  an  integer,  how  many  places  do 
you  point  off  for  decimals  in  the  quotient  ? 

282.    Examples  for  the   Slate. 

83.  Divide  6.24  by  12.  SQ.   Divide  1.576  by  8. 

84.  Divide  288.9  by  9.  87.    Divide  112.84  by  11. 
^B.   Divide  91.05  by  15.  ^%.   Divide  14.607  by  27. 


DIVISION  OF  DECIMALS.  113 

To  divide  Integers,  carrying  the  Division  to  Decimals. 

283.  e.   Find  1  fourth  of  13. 

wiaxTEx  "WGiiK.        In  Example  e,  after  the  integer  is  divided  there 
4)  13.00  is  a  remainder  of  1.     This  we  change  to  10  tenths. 

3  25  One  fourth  of  10  tenths  is  2  tenths,  and  2  tenths  re- 

main, which  are  equal  to  20  hundredths.   One  fourth 
of  20  hundredths  is  5  hundredths.     The  entire  quotient  is  3.25. 

Ill  tlio  following  examples  carry  the  division  to  decimals : 

89.  Find  1  fifth  of  157.  91.   Find  1  fourth  of  $  927. 

90.  Find  1  sixth  of  879.  92.    Find  1  eighth  of  §  721. 

93.  If  8  hamniochs  cost  $  7,  wdiat  is  the  cost  of  1  liammock  ? 

94.  If  4  dozen  hats  cost  $  78,  what  is  the  cost  of  1  hat  ? 

284.  /.   Divide  348  by  71. 

WKiTTEN  woKK.  We  divide  as  in  Example  e  till  we  have 

11)  348  f4  901  thousandths  in  the  quotient  and  there  is  still 

noj^  a  remainder.     We  might  keep  on  dividing, 

and  thus  come  to  greater  accuracy  in   the 

^'^O  quotient.     But  it  is  not  generally  necessary 

C39  to  carry  the  division  beyond  thousandths. 

-^  In  the  following  examples  carry  the 

division  to  thousandths,  and  if  there  is 

^^  a  remainder  then,  indicate  that  there  is 

one  by  dots,  as  in  the  answ^er  to  Example  /. 

95.  How  many  are  4321-9?  98.   Divide  876.1  by  8. 

96.  How  many  are  3214-12?         99.    Divide  13.47  by  17. 

97.  How  many  are  9486-19?       100.   Divide  9841  by  21. 

To  divide  an   Integer  or  a  Decimal  by  a  Decimal. 

285.  g.   How  many  times  are  0.3  contained  in  0.6  ?   0.4  in 
0.8?   0.6  in  1.2?   0.9  in  2.7  ? 

72.  How  many  times  are  0.03  contained  in  0.06?   0.08  m 
^.16?   0.12  in  1.44?  ^^'"'7'^- 

i.   Divide  0.006  by  0.002  ;   0.018  by  0.003;   flfti^ty  0.O12.- 


Uiri7E 


114  DIVISION  OF  DECIMALS. 

286.  J.   Divide  35  by  0.7.     k.  1.44  by  0.8. 

WRITTEN  WORK.  Before  dividing  by  a  fraction,  the 

(j.)  {k.)  dividend  must   be   expressed   in  the 

0.7)  35.0a        0.8)  1.4a4  same  denomination  as  the  divisor.    In 

^  H  o  Example  j,  the  divisor  is  a  number 

of  tenths  ;   the  dividend  expressed  in 

tenths  is  35.0  (350  tenths).     350  tenths  divided  by  7  tenths  gives  the 

same  quotient  as  350  divided  by  7,  which  is  50.  Ans.  50. 

In  Example  k,  the  divisor  is,  a  number  of  tenths  ;  the  dividend 

expressed  in  tenths  is  14.4  tenths  (the  denomination  may  be  indicated, 

as  in  the  written  work,  by  a  caret).     14.4  tenths  divided  by  8  tenths 

gives  the  same  quotient  as  14.4  divided  by  8,  which  is  1.8.     Ans.  1.8. 

In  Example  j,  bow  did  you  prepare  the  dividend  to  divide  ? 
How  did'  you  then  divide  ? 

In  Example  k,  bow  did  you  prepare  the  dividend  to  divide  ? 
How  did  you  indicate  the  denomination  tenths  in  the  dividend? 

287.  To  divide  by  decimals  :  1.  Uxpress  the  dividend  in 
the  same  denomination  as  the  divisor  hy  putting  a  mark 
as  many  places  to  the  right  of  the  decimal  point  as  there 
are  decimal  places  in  the  divisor. 

2.  Divide  as  if  the  divisor  were  an  integer y  fixing  the 
decimal  point  of  the  qiiotient  when  the  terms  of  the  divi- 
dend have  been  used  as  far  as  the  mark. 

288.     Examples  for  the  Slate. 

101.  Divide  315  by  0.7.  106.  Divide  1000  by  0.001. 

102.  Divide  86.1  by  0.08.  107.  Divide  90.09  by  0.071. 

103.  Divide  70.32  by  0.38.  108.  Divide  8.64  by  71.6. 

104.  Divide  0.172  by  0.12.  109.  Divide  538.1  by  4.001. 

105.  Divide  0.5307  by  0.1.  110.  Divide  3.027  by  21.1. 

111.  How  many  tops  at  $  0.02  apiece  can  be  bought  for  $  2  ? 

112.  How  many  rods,  each  16.5  feet,  are  there  in  100  feet  ? 

113.  At  $  0.125  a  quart  for  cherries,  how  many  quarts  can 
he  bought  for  $  0.25  ?   for  $0,875?   for  $1.25?   for  $3? 

For  other  examples  in  Multiplication  and  Division,  see  page  129.  — 

J 


PERCENTAGE.  115 


SEOTIOS"   X. 

PERCENTAGE. 

Oral  Examples. 

289.  a.  One  is  what  part  of  2  ?  of  3?   of  4?  of  5?  of  10? 

b.  One  is  wha^  part  of  20  ?   of  30  ?   of  50  ?   of  100  ? 

c.  What  part  of  100  is  1  ?   is  2?   3?   4?   5?   10?   20? 

Suggestion.    One  is  -^^  of  a  hundred;  2  is  yfg-;   3  is  yf^;  and  so  on. 

290.  Any  number  of  hundredths  of  a  thing  or  number 
is  a  per  cent  of  that  thing  or  number. 

Thus,  y§^  of  a  number  is  2  per  cent  of  it. 

d.  What  per  cent  of  a  number  is  y  §^  of  it  ?  y§^  ?  -j%%  ?  \%%  ? 

e.  To  take  20  per  cent  of  a  number  is  to  take  how  many 
hundredths  of  the  number  ? 

/.    How  many  hundredths  of  a  number  is  10  per  cent  of  it  ? 
25  per  cent  of  it  ? 

To  express  a  given  Per  Cent. 

291.  The  sign  %  is  used  for  the  words  ''  per  cent." 
Thus,  3  %  means  3  per  cent. 

292.  Any  per  cent  may  be  expressed  as  a  common  frac- 
tion, as  a  decimal,  or  with  the  sign  for  per  cent,  %.    Thus, 

1    per  cent  may  be  expressed  y^,  0.01,  orl%. 

5    percent    "  "  y|^,  0.05,  or  5%. 

8J  per  cent    "  "  ^,  0.08^,  or  8^%. 

100    percent    "  "  1^,  1.00,  or  100%. 

125    percent    "  "  |||,  1.25,  or  125%. 

4  per  cent    "  "  y^,  0.00^,  ori%. 

293.     Examples  for  the  Slate. 

Express  the  following  in  the  three  forms  given  above  : 
(1.)  2  per  cent.  (3.)  b\  per  cent.  (5.)  145  per  cent. 

(2.)  8  per  cent.  (4.)  100  per  cent.  (6.)  ^  per  cent. 


116  PERCENTAGE. 

To  change  a  Common  Fraction  to  a  Per  Cent. 

294,   a.  What  per  cent  of  a  number  is  J^  of  it  ?     &.  ^  of  it  ? 

"vvniTTEN  woKK.  (^a.)   Since  any  number  equals  ICO  per 

(a.)  {b.)  cent  of  itself,  \  of  the  number  must  equal 

2)  100  %  3)  100  %        \  of  100  % ,  or  50  % .  Ans.  50  % . 

50%  331%  (ft.)   -J  of  the  number  must  equal  \  of 

100  %  J  01^  33  J  % .  Ans.  33  J  % . 

295.     Oral  Examples. 
You  will  need  to  practise  on  these  exercises  till  you  can  give 
the  answers  at  sight. 

c.  What  per  cent  of  a  number  is  -J  of  it  ?   ^  ?    ^^  ?  t^^? 

d.  What  per  cent  of  a  number  is  5  of  it  ?    f  ?    f  ?    J  ?    |  ? 

A  9     3  9     49     7  9      3    9       7    9      0    9 
IT  •      8   •      ^  •      8   •      T(J  •      To  •,     1(7  • 

e.  What  per  cent  of  a  number  is  ^^  of  it  ?  /^  ?  ^f^  ?  ^5  ? 
^^  9    ^T_  9      7   9 

293.  From  what  you  have  learned  in  the  preceding 
exercises,  you  may  tell  what  part  of  a  number  the  following 
per  cents  are,  giving  the  parts  in  their  smallest  terms,  thus, 
"10%,  T^t,;   20V.,  l?  etc. 

f.  10%;  20%;  30%;  50%;  25%;  75%;  80%;  90%;  m%; 
37i%;  621%;  87^%;  33i%;  660o',  16S%. 

To    find    a  Number   that   is   a   given   Per   Cent   of  another 

Number. 

Oral  Examples. 
297.    g.  What  number  is  7%  of  S800  ? 
Solution.  —  7  %  of  $  800  is  ^^  of  §  800,  or  $  56.  Ans.  $56. 

A  number  found  by  taking  a  number  of  hundredths  of 
another  number  is  a  percentage  of  that  number.  Thus, 
in  the  example  above,  S  50  is  a  percentage  of  S  800. 

h.  What  is  4%  of  300  men  ?   of  S  500  ?   of  1000  ? 

i.    What  is  10%  of  8  50?   of  $25?   of  $  80  ? 


WniTTEX 

WORK. 

(a.) 

(ft.) 

946 

87.G0 

0.15 

0.08^ 

PERCENTAGE.  117 

J.   What  is  1%  of  2000  ?  3%  ?  5%  ?  6]%  ? 

A:.  What  is  1%  of  $1200  ?  i%  ?    1%  ?    i%  ?   2%  ?   2^%  ? 

i.    What  is  1%  of  $800?   100%  ?   101%?   102%  ?   110%  ? 

122.  A  farmer  raised  400  bushels  of  corn,  and  sold  50%  of  it. 
How  many  bushels  did  he  sell  ? 

n.  In  a  school  of  boys  and  girls  containing  300  pupils,  70  % 
were  girls.     How  many  were  boys  ? 

298.    Examples  for  the  Slate. 

a.  What  is  15%  of  $  946  ?     b.  SJ  %  of  $  87.60  ? 

By  what  do  you  divide  a  number  to 
findl%  of  it? 

Having  found  1%,  how  do  you  find 
"4730  29 vn'       ^^^  number  of  times  1  %  ? 

946  70080  Does  it  make  any  difference  which 

141  90  7.3000         operation  you  perform  first  ? 

(7.)  What  is  6%  of  $469?       (14.)  J%  of  120  acres  ? 
(8  )  8%  of  975  ?  (15.)  §%  of  784  pounds  "" 

(9.)  9%  of  1800  ?  (16.)  ^5%  of  1600  days  ? 

(10.)  16%  of  465?  (17.)  t%  of  750  words? 

(11.)  25%  of  6800  men  ?  (18.)  40%  of  7.45  ? 

(12.)  42%  of  876  miles  ?  (19.)  60%  of  10.5  tons  ? 

(13.)  1^1^^%  of  873  bushels.        (20.)  18^%  of  960  barrels  ? 

21.  Of  a  flock  of  175  sheep  4%  were  killed  by  dogs.  How 
many  were  killed  ? 

22.  A  stable-keeper  bought  two  carriages ;  for  the  first  he 
paid  $275,  and  for  the  second  75%  as  much  as  for  the  first. 
What  did  he  pay  for  both  ? 

23.  A  landlord  has  lowered  his  rents  20%.  What  will  he 
now  charge  for  tenements  that  have  rented  for  $  200  a  year  ? 
for  $  425  ?   for  $  12.50  a  month  ? 

24.  What  does  a  man  receive  for  a  house-lot  which  cost 
$1524  by  selling  it  for  81%  more  than  it  cost  ? 

2o.  What  prices  must  be  charged  for  four  kinds  of  tea  cost- 
ing 36/,  48/,  54/,  and  63/  per  pound  to  gain  16§%  ? 


118  PERCENTAGE. 

To  find  a  Number  when  a  Per  Cent  of  it  is  given. 

Oral  Examples. 

299.    a.    20%  of  a  number  is  $40.     What  is  the  number  ? 

Solution.  —  Since  20  %  of  the  number  sought  is  $  40,  1  %  of  the 
number  sought  is  -^^  of  $40,  which  is  $2,  and  100%  of  the  number 
sought  is  100  times  $2,  which  is  $200.  Arts.  $200. 

h.    60  pounds  is  6  %  of  how  many  pounds  ? 

c.  $24  is  8%  of  how  many  dollars  ?  $  30  is  12%  of  how 
many  dollars  ? 

d.  90  is  10  %  of  what  number  ?    220  is  11  %  of  what  number  ? 

e.  120  bushels  is  60%  of  what  number?  $80  is  40%  of 
what  number  ? 

/.  John  has  12/,  which  is  6%  of  what  Charles  has.  How 
much  has  Charles  ? 

g.  A  man  gained  $  20  by  selling  his  watch  for  10  %  more 
than  it  cost  him.     How  much  did  it  cost  ? 

h.  Mary  spelt  98  %  of  the  words  given  out  to  her  class  and 
missed  the  rest,  which  was  6  words.  What  per  cent  did  she 
miss  ?     How  many  words  were  given  out  ? 

300.    Examples  for  the  Slate. 

a.   $  250.50  is  25%  of  what  number  ? 

WRITTEN  WORK.  How   do  you  find  1%   of  a  number 

r}Jl^2i     -.oA  when  a  number  of  per  cents  are  given? 

— 1UU.U.  When  you  have  found  1  %  of  a  num- 

Ans  $  1002.         ^^^y  ^^^  ^^  y^^  ^^^  ^^^  whole  number  ? 
(26.)  74  is  37%  of  what  ?         (30.)  $  75  is  37|%  of  what  ? 
(27)  150.50  is  5%  of  what  ?    (31.)  1 94.50  is  6J%  of  what  ? 
(28.)  $  4.86  is  10  %  of  what  ?    (32.)  $31.25  is  125  %  of  what  ? 
(29.)  $  649.33  is  11  %  of  what  ?  (33.)  $  87.50  is  62  J  %  of  what  ? 

34.  A  man  expends  70  %  of  his  income  and  saves  $  369. 
What  is  his  income  ? 

35.  I  drew  from  a  bank  $  1485,  which  was  45%  of  what  was 
left.     What  w^as  the  sum  in  at  first  ? 


PERCENTAGE.  119 

To  find  what  Per  Cent  one  Number  is  of  another. 

Oral  Examples. 
301.  a.  What  per  cent  of  50  is  7  ? 
Solution.  —  7  is  -^-^  of  50.     -^^j-  equals  ^^q,  or  14%.  Am.  14%. 

b.  What  per  cent  of  20  is  1  ?    is  3  ?    is  9  ? 

c.  What  per  cent  of  10  is  1  ?    is  3  ?   is  7? 

d.  What  per  cent  of  25  is  2  ?   is  3  ?   is  6  ? 

e.  What  per  cent  of  80  is  20  ? 

Solution.  —  20  is  i-  of  80.     J  equals  ^,  or  25  % .  Ans.  25  % . 

/.  What  per  cent  of  40  is  8  ?  of  16  is  4  ?  of  12  is  6  ?  of  25 
is  5? 

g.  A  boy  had  20  plums,  and  gave  away  7  of  them.  What 
per  cent  did  he  give  away  ?     What  per  cent  did  he  keep  ? 

h.  A  man  hired  $  50,  and  paid  $  3  for  the  use  of  it.  What 
per  cent  did  he  pay  ? 

i.  A  boy  who  weighed  80  pounds  lost  10  pounds  by  sickness. 
What  per  cent  did  he  lose  ? 

J.  A  watch  which  cost  $  20  was  sold  for  $  15.  How  many 
dollars  were  lost  ?     What  per  cent  was  lost  ? 

302.    Examples   for  the   Slate. 

k.  What  per  cent  of  $  50  is  $  3.25  ? 

WRITTEN  WORK.  To  find  what  per  cent  one  mim- 

50)  3.25  (0.06f  j-6i%.     ber  is  of  another,  Divide  the  per- 

^  ^^  centage  hy  the  number  it  is  a  per- 

25  centage  of,  carrying  the  division  to 

hundredths. 

36.  What  per  cent  of  $  480  is  $  40  ? 

37.  Of  16  days  is  14  days  ?         39.    Of  15  tons  is  2  tons  ? 

38.  Of  35  cents  is  8  cents  ?         40.    Of  1  score  is  1  dozen  ? 

41.  If  I  sell  a  horse  for  $  100  which  cost  $  150,  what  per 
cent  do  I  lose  ? 

42.  Flour  which  was  bought  at  $  4.80  per  barrel  is  sold  at 
$  6.00.     What  is  the  per  cent  of  gain  ? 


120  PROFIT  AND  LOSS. 

PEOriT  AND  LOSS. 
Oral  Examples. 

303.  a.  How  much  money  is  gained  hy  selling  a  cow  for 
10%  above  cost,  the  cost  being  $  60  ?   $80  ? 

b.  How  much  money  is  lost  on  a  horse  which  cost  8  200  by 
selling  liim  at  a  loss  of  10%  ?    of  20%  ?   of  50%? 

c.  At  what  price  must  books  which  cost  $4  apiece  bo  sold 
to  gain  50%?   25%?   20%?    10%? 

d.  A  man  bought  butter  at  20  cents  a  pound.  At  what 
price  must  he  sell  it  to  lose  25%  ?    20%  ?    30%  ? 

e.  A  man  sold  a  cow  for  §  GG,  and  gained  10%.  What  did 
she  cost  him  ? 

Solufion.  — Since  he  gained  10%,  $66  must  be  110%  of  the  cost. 
If  S  66  is  1 J^  or  \^  of  the  cost,  J-^  of  the  cost  must  be  ^^  of  $  66,  or  $6, 
and  the  whole  cost  must  be  10  times  $6,  or  §60.  Ans.  $  60. 

/.  What  must  have  been  paid  a  bushel  for  wheat,  if  by  sell- 
ing it  for  §1.50  there  was  a  gain  of  25%  ?    of  50%  ? 

g.  A  man  sold  a  wagon  for  $48,  and  lost  20%.  What  was 
the  cost  of  the  wagon  ? 

Suggestion.     Since  he  lost  20%,  §48  is  80%,  or  |-  of  the  cost. 

h.  What  was  the  cost  of  knives  which  sold  for  27  cents 
each  at  a  loss  of  10%?   25%? 

2.  Wliat  per  cent  would  be  gained  by  selling  a  plough  for 
$10  which  cost  $8? 

Suggestion.   §  2  was  gained  on  §  8,  which  was  |-  or  \-  of  8.    ^  =  25  %. 

j.  WHiat  per  cent  is  gained  if  spades  costing  80  cents  are 
sold  for  $  1  00  ?   for  §  1.20  ?   for  $  1.30  ? 

304.  The  difference  between  the  cost  of  goods  and  the 
price  at  which  they  are  sold  is  a  profit  or  loss. 

305.     Examples  for  the  Slate. 

43.  Cloth  Avhich  was  bought  at  $  7  a  ^^ard  was  sold  at  a  gain 
of  12%.     W^liat  was  received  for  it  ? 

44.  Por  wliat  must  I  sell  150  cords  of  wood  which  cost  $  3.50 
a  cord  to  gain  8%  ? 


COMMISSION,  INSURANCE,   TAXES,  ETC.  121 

45.  By  selling  corn  at  25  %  profit  a  merchant  gained 
$462.50.     Wliat  did  the  corn  cost  him? 

46.  From  a  barrel  of  kerosene  15 1^  gallons  leaked  out.  If 
this  was  30%  of  the  contents  of  the  barrel,  how  many  gallons 
did  the  barrel  contain  at  first  ? 

47.  Tlie  expenses  of  a  family  in  1877  were  %B^^.  In  1878 
they  were  7?i%  less.     Wliat  were  their  expenses  in  1878  ? 

48.  A  shoe-dealer  sold  boots  at  8  4.50  a  pair  at  a  gain  of 
20%  on  wliat  they  cost  him.     What  did  tliey  cost? 

49.  If  120%  of  a  number  is  $  4.50,  wliat  is  the  number  ? 

50.  A  man  sold  coal  at  $5.50,  and  lost  23%  of  the  cost. 
What  did  his  coal  cost  ? 

51.  What  per  cent  is  gained  by  selling  nutmegs  at  56  cents 
a  pound  which  cost  48  cents  a  pound  ? 

52.  A,  having  failed,  pays  B  $  280  instead  of  $  480,  which 
he  owed  him.     What  per  cent  does  B  lose  ? 

306.    OOMMISSIGl^,  lETSUEANOE,  TAXES,  ETO. 

Note.  The  tenns  Commission,  Insurance,  Taxes,  etc.,  should  be  ex- 
plained by  tlie  teacher.  A  full  treatment  of  these  subjects  will  be  found  in 
the  Fi-anklin  Written  Arithmetic. 

53.  An  agent  sold  a  house  for  $  2500.  If  he  received  a  com- 
mission of  4%  for  selling,  how  much  did  he  receive  ? 

54.  A  lawyer  received  5%  commission  for  collecting  a  bill  of 
$  375.     What  was  his  commission  ? 

bo.  An  architect  received  a  commission  of  §  140.30  for  plan- 
ning and  superintending  the  building  of  a  house.  If  his  com- 
mission was  3%,  what  was  the  cost  of  the  house? 

56.  A  man  had  his  liousc  insured  for  §2000.  If  he  paid 
1J%  of  this  sum  for  the  insurance,  what  did  he  pay  ? 

57.  A  man  is  taxed  for  property  amounting  to  $  4760.  If 
his  tax  is  1]  %  of  tin's  amount,  what  is  liis  tax  ? 

lj?>.  A  broker  bought  for  a  man  5  shares  of  railroad  stock 
wortli  8 100  a  share,  and  charged  J  %  connnission.  What  was 
Ilia  charge  ? 


122  INTEREST. 

SECTION   XI. 

INTEREST. 

307.  A  man  lent  me  $200  for  1  year.  At  the  end  of  the 
year  I  paid  him  back  $  200,  with  5  %  of  %  200  for  the  use  of 
the  money.  How  much  did  I  pay  for  the  use  of  the  money? 
How  much  in  all  ?         Ans.  $  10  for  the  use  ;   $  210  in  all. 

Money  paid  for  the  use  of  money  is  interest.  The 
money  for  the  use  of  which  interest  is  paid  is  the  principal. 

The  sum  of  the  principal  and  interest  is  the  amount. 

Thus,  in  the  above  example  $  10  is  the  interest,  $  200  is  the 
principal,  and  $  210  is  the  amount. 

308.  To  find  the  interest,  a  certain  per  cent  of  the  prin- 
cipal is  taken  for  each  year.  This  per  cent  is  called  the 
rate  per  cent,  or  simply  the  rate. 

Note.  In  reckoning  interest,  it  is  customary  to  consider  a  year  to  be 
12  months,  and  a  month  30  days. 

309.  a.  What  is  the  interest  of  $200  for  2  years  at  6%  ? 

WKITTEN   WORK. 

200  6%  is  ^fo,  or  0.06.      0.06  of  $200  is  $12,  the 

0.06  interest  for  1  year. 

12.00  The  interest  for  2  years  is  2  times  $  12,  or  $  24. 

2  Ans.  $24. 


24.00  310.     Oral  Examples. 

What  is  the  interest 

b.  Of  $100  at  6%  for  2  years  ?    3  years  ? 

c.  Of  $200  for  2  years  at  3%  ?   4%  ?   5%  ?   7%?   12%? 

d.  Of  $40  at  10%  for  1  year?    for  3  years  6  months  ? 

SOLUTION. 

1 40  X  10%  =  $ 4.   Int.  for  1  y.  (year).  3  years  6  months 

$4  X  3i     -  $  14.    "      "   3  y.  6  mo.  (months),     equals  3^  years. 


INTEREST.  123 

What  is  the  interest  of  160  at  10% 

e.  For  1  y.  6  mo.  ?  g.   For  1  y.  9  mo.  ? 

/.    For  1  y.  3  mo.  ?  h.  For  2  y.  8  mo.  ? 

What  is  the  interest  of  $120  at  10% 

i.    For4y.  2mo.  ?  -6:.   For  2  y.  5  mo.  ? 

7.    For  4  y.  1  mo.  ?  L   For  6  y.  7  mo.  ? 

The  Six  Per  Cent  Method. 

A.  general  method  of  computing  interest  will  be  found  on  p.  \42  of  the  Appendix. 
311.      Oral  Examples. 

a.  At  6%,  how  many  hundredths  of  the  principal  is  the  in- 
terest for  1  y.  ?    for  2  y.  ?   for  3  y,  ?     Arts.  0.06 ;    0.12  ;   0.18. 

At  6%,  how  many  hundredths  of  the  principal  is  the  interest 

b.  For  6  y.  ?   for  8  y.  ?    10  y.  ?    11  y.  ? 

c.  For  6  mo.  ?    for  2  mo.  ? 

Suggestion.  Since  the  interest  is  6  %  for  1  year,  for  6  months,  which 
is  1  half  of  a  year,  it  is  3  % ,  or  0.03  ;  for  2  months,  which  is  1  sixth  ot 
a  year,  it  is  1  %,  or  0.01  ;  and  so  on. 

At  6%,  how  many  hundredths  of  the  principal  is  the  interest 

d.  For  2  mo.  ?   4  mo.  ?   8  mo.  ?    10  mo.  ? 

e.  For  1  y.  6  mo.  ?  2  y.  6  mo.  ?  3  y.  2  mo.  ? 
/.  For  4  y.  2  mo.  ?  3  y.  4  mo.  ?  5  y.  8  mo.  ? 
g.    For  1  mo.  ?   3  mo.  ?    5  mo.  ? 

Suggestion.  Since  the  interest  is  1  %  for  2  months,  for  1  month  it 
is  •^%,  or  0.005,-   for  3  months  it  is  1^%,  or  0.015  ;  and  so  on. 

At  6%,  what  part  of  the  principal  is  the  interest 

h.    For  3  mo.  ?    5  mo.  ?    7  mo.  ?    9  mo.  ?    11  mo.  ? 

i.    For  1  y.  1  mo.  ?    2  y.  1  mo.  ?    2  y.  3  mo.  ?    2  y.  2  mo.  ? 

j.    For  3  y.  1  mo.  ?   4  y.  5  mo.  ?   3  y.  6  mo.  ?   3  y.  7  mo.  ? 

k.  For  5  y.  10  mo.  ?    6  y.  9  mo.  ?    6  y.  8  mo.  ?    10 y.  11  mo.? 

1.    For  6  days  ?    for  1  day  ?   for  5  days  ?   for  7  days  ? 

Suggestion.  Since  the  interest  is  0.005  of  the  principal  for  1  month 
or  30  days,  for  6  days,  whf-^h  is  1  fifth  of  a  month,  it  is  0.001  ;  for  1 
day  it  is  0.000^  ;  for  5  days,  O.OOOf  ;   for  7  days,  0.001^;  and  so  on. 


124  INTEREST, 

At  6%,  what  part  of  the  principal  is  the  interest 

m.  For  6  d.  (days)  ?   12  d.  ?   18  d.  ?   24  d.  ?   1  mo.  (30  d.)  ? 

n.  For  Id.?   3d.?   9d.  ?   10  d.  ?   lid.?   13  d.? 

o.  For  15  d.?   19  d.?   25  d.  ?   17  d.?   21  d.?   27  d.  ? 

312.     Examples  for  the  Slate. 

a.   At  6%,  what  part  of  the  principal  is  the  interest  for  2  y. 
5  mo.  14  d.  ? 

WRITTEN  WORK.  To  fiiid  wtat   part  of  the 

0.12      for  2  years.  principal  the  interest  is,  r«Z;e 

0.025      "   5  months.  ^  ^^-^^^^^  ^^^  ^^        hundredths  as 

0.002 \    "   14  days.  ,7  .  .    ,. 

^^  ^  ^  ^  '^^    ^^  ^      ^^       ^  ^  ^         m(j?'e  a?'^  ?/cars,  1  hcdj  as  many 

hundredths  as  there  are  months, 


0.147i    "  2  y.  5  mo.  14  d. 


and  1  sixth  as  many  thousandths  as  there  a?r,  days. 

At  6%,  what  part  of  the  principal  is  the  interest 

1.  For  1  y.  1  mo.  Id.?  6.   For  4  y.  5  mo.  25  d.  ? 

2.  For  1  y.  3  mo.  5  d.  ?  7.   For  8  y.  4  mo.  ? 

3.  For  2  y.  6  mo.  9  d.  ?  8.   For  6  mo.  tO  d.  ? 

4.  For  3  y.  8  mo.  23  d.  ?  9.    For  2  y.  10  d,  ? 

5.  For  5  y.  7  mo.  7  d.  ?  10.    For  3  y.  5  /no.  14  d.  ? 

313.   b.  Find  the  interest  of  $120  at  6%  for  3  y.  5  mo.  14  d. 

WEiTTEN  WORK.  At  6  % ,  the  interest  for  3  years  5  monUis  14  days 

120  IS  0.207^  of  the  principal.    0.207-^  of  $  120  is  §  24.88. 

0  2071  Ans.  $24.88. 

7^  To  compute  the  interest  on  any  sum  at 

840  ^  %  '-  (I)   Find  the  decimal  that  expresses  the 

240  pcirt  ivhich  the  iyitercst  is  of  the  ^9?v!7irzpaZ ; 


24.880  (2)  iy  t^^is  decimal  midtiply  the  lorinci'pal. 

Note.     In  doing  the  following  examjoles,  get  the  answers  to  the  nearest 
cent. 

At  G%,  what  is  the  interest 

11.  Of  $  800  for  1  y.  4  mo.  ?     13.  Of  $  350  for  3  y.  G  mo.  9  d.  ? 

12.  Of  $  124  for  2  y.  7  mo.  ?     14.  Of  $  488  for  5  y.  7  mo.  14  d.  ? 


INTEREST.  125 

314.  15.  At  6%,  T\'liat  is  the  interest  of  $500  from  Jan.  1, 
1877,  to  March  7,  1878  ? 

Suggestion.  From  Jan.  1,  1877,  to  Jan.  1,  1878,  is  1  year ;  from 
Jan.  1  to  March  1  is  2  months  ;  and  from  March  1  to  March  7  is 
6  (lays.     The  whole  time  is  1  y.  2  mo.  G  d. 

At  G%,  ^vhat  is  tlie  interest 

16.  Of  §  268.90  from  July  15,  1877,  to  Sept.  25,  1878  ? 

17.  Of  §  1185  from  May  3,  1878,  to  August  18,  1879  ? 

18.  Of  $  2000  from  Oct.  25,  1878,  to  Jan.  29,  1879  ? 
Sitggsstion.     The  whole  time  is  3  months  4  days. 

At  6%,  what  is  the  interest 

19.  Of  S  87.50  from  Sept.  14,  1877,  to  Feb.  20,  1878  ? 

20.  Of  8380.70  from  Sept.  17,  1877,  to  Marcli  29,  1879  ? 

21.  Of  S  200  from  Dec.  16,  1877,  to  May  10,  1878  ? 
Suggestion.     From  Dec.  16,  1877,  to  Ajoril  IG,  1878,  is  4  months. 

From  April  16  to  April  30  is  14  days,  and  to  May  10  is  10  days  more, 
or  24  days.     The  whole  time  is  4  months  24  days. 

At  6%,  w^hat  is  the  interest 

22.  Of  $  95  from  Nov.  26,  1878,  to  May  15,  1879  ? 

23.  Of  §284.40  from  Nov.  27,  1877,  to  Oct.  15,  1879? 

24.  At  6%,  what  is  the  amount  of  $  7500  from  Feb.  15, 1878, 
to  Dec.  29,  1879  ? 

Suggestion.     To  find  the  amount,  add  the  principal  to  the  interest. 
At  6%,  wdiat  is  the  amount 

25.  Of  $  496  from  June  17,  1877,  to  Dec.  1,  1878  ? 

26.  Of  $  630.20  from  Feb.  18,  1877,  to  July  5,  1878  ? 

315.     To  find  the  Interest  at  any  Per  Cent  other  than  6%. 

What  is  the  interest  of  8  2S5,  for  3  y.  4  mo. 

27.  At7%?  28.   At8%?  29.   At  5%  ? 
Suggestion.     First  find  the  interest  at  6%  ;  then  to  find  the  interest  at 

7%,  add  1  of  this  interest  to  itself,  to  find  the  interest  at  8%,  arZr/  |  or  J. 
To  find  the  interest  ai  5%,  subtract  J,  and  so  on. 


126  INTEREST. 

What  is  the  interest  of  $  156.60  for  3  y.  6  mo.  18  d. 
30.  At  4%  ?  31.   At  8%  ?  32.   At  10%  ? 

What  is  the  interest 

33.  Of  $  474  for  2  y.  11  mo.  at  7  %  ? 

34.  Of  $2500  from  Oct.  15, 1878,  to  July  27, 1879,  at  10%  ? 
What  is  the  interest  and  amount 

35.  Of  $  318  for  11  mo.  90  d.  at  5%  ? 

36.  Of  1 268  for  3  y.  8  mo.  27  d.  at  8  %  ? 

37.  Of  $  98.25  for  1  y.  5  mo.  1  d.  at  7%  ? 
What  is  the  amount 

38.  Of  $  805.50  from  Sept.  10, 1878,  to  Nov.  4, 1878,  at  6^%  ? 

39.  Of  $ 962.25  from  Apr.  8,  1875,  to  Oct.  1,  1875,  at  7%  ? 

40.  Of  $  500  from  Jan.  1,  1878,  to  Jan.  15,  1879,  at  7^%  ? 

41.  A  note  for  $400  was  on  interest  at  6%,  from  June  5, 
1875,  to  June  5,  1876,  when  a  payment  of  $  124  was  made  to 
pay  the  interest  due  and  part  pay  the  note.  How  much  then 
remained  due  ? 

42.  If  $  300  was  due  on  a  note  June  5,  1876,  and  was  kept 
on  interest  till  December  5,  1876,  what  was  then  required  to 
pay  the  note  and  the  interest  due  ? 

BANK  DISCOUNT. 
316.    On  the  7th  of  June,  1878,  Mr.  M.  J.  Oliver  bought 
a  horse  of  Mr.  John  Day,  and  agreed  to  pay  $  225  for  it  in 
60  days,  giving  him  the  following  written  promise,  called  a 

PROMISSORY   NOTE. 

^ ^2^  X«icoZ?i,  June  7,  18118, 

(25^^  c/ayd  /i^7ri  c/ci^  of^ ^'toTncae  '^  ^uiy  ^oupi.  .^czy. 

If  Mr.  Day  wants  the  money  before  the  60  days  are  over, 
he  can  take  the  note  to  a  bank  and  get  the  money  by  pay- 


BANK  DISCOUNT.  127 

ing  the  bank  the  interest  on  it  for  the  time  to  elapse  before 
it  is  due,  and  for  3  days  more. 

The  note  is  then  said  to  be  discounted. 

317.  The  interest  paid  to  the  bank  is  bank  discount. 

318.  The  money  received  from  the  bank  is  called  the 
proceeds. 

319.  The  three  days  for  which  interest  is  taken  beyond 
the  time  named  in  the  note  are  called  days  of  grace. 

320.  a.    Find  the  discount  on  the  above  note  July  7,  1878, 
the  discount  being  6%.     Find  the  proceeds. 

WRITTEN  WORK.  Bank  discount  is  interest  for  the  specified  time 

$  225  ^^d  of  3  days'  grace. 

0.0055  '^^^  ^^^^^  horn  July  7  to  the  end  of  60  days  and 

—7^  grace  is  33  days. 

-1I9K  The  interest  of  1 225  for  33  days  at  6%  is  $  1.24, 

1125  which    is   the   discount.      $225  less   $1.24  equals 


$  1.2375  $  223.76,  the  proceeds  of  the  note. 

$  225  -  $  1.24=$  223.76.        ^^s-  $  1-24,  discount ;  $  223.76,  proceeds. 

321.    Examples  for  the  Slate. 

43.  What  is  the  hank  discount  of  a  note  for  $  500,  payable 
in  30  days,  discount  6  % .     What  are  the  proceeds  ? 

Note.     Add  3  days  of  grace  to  the  specified  time  in  all  instances. 
Find  the  bank  discount  and  proceeds  of  a  note 

44.  For  $300,  payable  in  60  days,  discount'6%. 

45.  For  175.80,  payable  in  90  days,  discount  7%. 

46.  For  $1300,  payable  in  4  months,  discount  5%. 

47.  For  $486.50,  payable  in  45  days,  discount  8%. 

48.  For  $275.48,  payable  in  3  months,  discount  10%. 

49.  For  $80,  due  July  15,  and  discounted  June  15,  at  6%. 

50.  For  $700,  due  Aug.  29,  and  discounted  Aug.  2,  at  5%. 

51.  For  $650,  dated  Aug.  25,  payable  in  2  months,  a^nd 
discounted  Sept.  25,  at  6%. 

52.  For  $2000,  dated  Oct.  1,  payable  in  1  month,  and  dis- 
counted Oct.  16,  at  9%. 


128 


DRILL   TABLE. 


322.    DRILL  TABLE  No.  4. 


Common  Fractions.  —  Decimals.  —  Percentage. 


A 

B 

c 

i 

4 

U 

4 

§ 

U 

i 

tV 

u 

\% 

rAr 

7 

Ha 

T^3 

1% 

i 

A 

is 

f 

tV 

iff 

H 

A 

H 

li 

3V 

fs 

f 

A 

«• 

§ 

# 

H 

0 

ifV 

40- 

T^d 

/a 

i-V 

* 

A 

J$ 

ft 

A 

-2S 

1 

tfj 

21 

5 

A 

ja 

i?" 

^ 

e 

-^^ 

5 

tf 

f 

A 

u 

H 

^4- 

i^/k 

3 

in 

f§ 

flT 

ttV 

T^A 

^ 

i§ 

§* 

? 

A 

il 

D 

E 

F 

3? 

5 

19J 

50 

IG 

12s 

125 

13 

ICtV 

8} 

19 

9j 

n 

.    7 

11§ 

12§ 

12 

155- 

Gf 

9 

8i 

14* 

18 

I'i 

n 

8 

12i 

iiA 

21 

17i 

52^. 

14 

8J 

9J 

8 

15J 

Cff 

17 

123- 

8* 

9 

14^ 

42 

15 

cj 

7J 

24 

llj 

5t\ 

11 

13j\ 

12$ 

13 

15-^ 

13i 

16 

20i 

11  sV 

7 

11^ 

162- 

22 

19? 

7i 

10 

n 

lOi 

12 

lltV 

8.^ 

25 

9* 

n 

13 

13i 

DRILL  EXERCISES. 


129 


DRILL  TABLE  No.  4 

{continued). 

Examples                 H  I 

1,  2.905  3.178 

2.  3.061  50.817 
0.052  0.703 

38.04  209.63 

200.1  54.68 

0.05  3.0721 

708.1  0.1001 
362  3  0.0732 

52.12  9.246 

1.805  43.287 

100.2  0.016 
0.086  9.03 

702.4  3.0072 

0.7  0.245 

5  099  3.18 

80.06  5.4061 

0.004  10.18 

581.9  3287 

600.7  540.91 

7.101  10.6 

71.01  4.8001 

19.06  13.704 

0.83  97  3001 

2^.5  2.007 

16.035  191.06 


Exercises  upon  the  Table. 

323.     Common  Fractions. 

124.  Change  C  to  smallest  terms. 

125.  Change  D  to  improper  fractions. 

126.  Add  A  and  B. 

127.  Add  A,  C,  and  D. 

128.  Add  D  and  F. 

129.  A-B.  136.  AxD. 

130.  G-F.  137.  DxF. 

131.  D-C.  138.  A-^E. 

132.  F-D.  139.  E-^B. 

133.  AxE.  140.  A-^B. 

134.  GxB.  141.  D-^A. 

135.  AxC.  142.  F^D. 

324.    Decimals. 

143.  Change  A  to  a  decimal  (3  places). 

144'  Change  B  to  a  decimal  (4  places). 

145.  Add  H,  I,  and  2.9784. 

146.  From  1000  take  H. 

147.  Find  the  difference  of  H  and  I. 

148.  Multiply  I  by  7. 

149.  Multiply  20  by  H. 

150.  Multiply  H  by  I. 

151.  Divide  H  by  4. 

152.  Divide  57  by  H  (3  places). 

153.  Divide  I  by  H  (4  places). 

325*     Percentage. 

154.  What  is  5  %  of  E  dollars  ? 

155.  What  is  7%  of  G  dollars? 

156.  What  is  8  %  of  H  dollars  ? 

157.  Find  the  interest  of  G  dollars  for 

3  y.  6  mo.  at  6  % . 

158.  Find  the  interest  of  E  hundred  dol- 

lars for  4  y.  9  mo.  1 5  d.  at  6  % . 

159.  Find  the  interest  of  G  hundred  dol- 

lars for  2  y.  3 mo.  14  d.  at  5  %. 


130 


MENSURATION, 


SECTION    XII. 


MENSURATION. 

Note.  For  those  who  may  study  no  liigher  Arithmetic,  brief  direction  -. 
are  here  given  for  finding  the  areas  of  some  surfaces  and  the  vohnne  of 
some  solids.  Further  exi)lanation3  and  illustrations  of  these  surfaces  and 
solids  should  le  given  by  the  teacher,  who  is  referred  to  the  Franklin 
Written  Arithmetic,  pages  137  to  139  ;  also  pages  277  to  288. 

SURFACES. 

326.  A  square  each  of  whose  sides  is 
1  inch  long  is  a  square  inch;  a  square 
each  of  whose  sides  is  1  foot  long  is  a 
square  foot;  etc. 

327.  The  area  of  a  surface  is  its  con- 
tents reckoned  in  square  units. 

A  Square. 

To  find  the  Area  of  a  Rectangle. 

328.    A  rectangle  is  a  four-sided  figure  which  has  all 
its  corners  square. 

a.  If  a  rectangle  is  4  inches  long  and 
1  inch  ^Yide,  how  many  square  inches  does 
it  contain  ?  How  many  will  it  contain  if 
it  is  4  inches  long  and  2  inches  wide  ?    3  inches  wide  ? 

b.  What  numbers  will  you  multiply 
together  to  find  the  number  of  square 
units  in  any  rectangle  ? 

329.  The  area  of  a  rectangle  is 
found  by  midtiplyincj  the  niivibcr  of 
units  in  the  length  hj  the  member  of 

like  units  in  the  hreadth.     This  is  expressed  for  brevity  as 

multiplying  the  length  hy  the  hreadth. 


A  Rectangle. 


MENSURATION.  131 

330.    Oral  Exercises. 

c.  How  many  square  inches  are  there  in  the  surface  of  a 
slate  that  is  9  inches  long  and  8  inches  wide  ?  that  is  12  inches 
(1  foot)  long  and  12  inches  (1  foot)  wide  ?  Then  how  many 
square  inches  are  there  in  a  square  foot  ? 

d.  How  many  square  feet  are  there  in  a  rectangle  that  is 
5  feet  long  and  3  feet  wide  ?  How  many  square  feet  are  there 
in  a  square  whose  sides  are  each  3  feet  (1  yard)  long  ? 

331.    Eepeat  the  following  tahle  : 

Square  Measure. 
144   square  inches        =1  square  foot. 
9    square  feet  =1  square  yard. 

m  square  yards  or  )       ^  ^  ^^^^  ^^^^ 
272i  square  feet  ) 

160    square  rods  =1  acre. 

640   acres  =1  square  mile. 

332,    Examples  for  the   Slate. 

1.  In  a  walk  20  feet  long  and  6  feet  wide,  how  many  square 
feet  ?  how  many  square  yards  ? 

2.  How  many  square  yards  of  carpeting  will  it  take  to  cover 
a  floor  that  is  15  feet  long  and  14  feet  wide  ? 

3.  In  a  piece  of  ground  200  feet  long  and  50  feet  wide,  how 
man}^  square  feet  ?   how  many  square  rods  ? 

4.  In  a  lot  250  feet  long  and  120  feet  wide,  how  many 
square  rods  ? 

5.  How  many  square  feet  of  land  in  a  lot  4  rods  wide  and 
90  feet  deep  ? 

6.  At  20/  a  square  foot  for  land,  what  must  be  paid  for  a 
lot  100  feet  deep  and  3 J  rods  on  the  front  ? 

7.  In  a  square  field  60  rods  on  each  side,  how  many  acres  ? 

8.  What  is  the  difference  in  the  size  of  two  lots,  one  con- 
taining %b  square  rods,  the  other  being  (j5  rods  square,  that  is, 
Qb  rods  long  and  Q6  rods  wide  ? 


132: 


MENSURATION. 


TRIANGLES,   POLYGONS,  AND  CIRCLES. 

333.   A  triangle  is  half  a  rectangle  of  the 
same  base  and  height.     Hence, 

To  find  the  area  of  a  triangle,  Multiply  the 
A  Triangle,     j^^^g  ly  fj^^  height  and  divide  the  product  hij  ^. 

334.  To  find  the  area  of  a  polygon,  Divide 
it  into  triangles  and  find  the  sicm  of  their  areas. 

335.  When  the  diameter  of 
a  circle  is  given, 

To  find  the  circumference, 
Multiply  the  diameter  hij  S.lJplS. 

336.  A  circle  may  be  said  to  consist  of 
triangles  whose  bases  form  the  circumfer- 
ence, the  height 

of  the  triangles  being  equal  to 
the  radius  of  the  circle.  Hence, 
To  find  the  area  of  a  circle. 
Multiply  the  circumference  hy  the 
radius  and  divide  the  product  hy  2. 


A  Polygon. 


A  Circle. 


337.    Examples  for  the  Slate. 

8.  I  have  a  piece  of  woodland  shaped  like  a  triangle,  the 
base  of  which  is  100  rods  long  and  the  height  62  rods.  How 
many  acres  does  it  contain  ? 

9.  How  many  square  yards  are  there  in  a  roof  that  is  made 
up  of  4  triangles,  the  base  of  each  being  20  feet  and  the  height 
15  feet  ? 

10.  I  have  a  tin-pail  that  measures  8  inches  across  the  top. 
What  is  the  distance  round  it  ?  How  many  square  inches  must 
there  be  in  a  cover  that  will  fit  it  ? 

11.  How  many  square  feet  of  land  in  a  circular  flower-bed 
that  measures  10  feet  across  ? 


MENSURATION. 


133 


SOLIDS. 

338.  A  rectangular  solid  is  a  solid  bounded  by  six 
rectangles.     A  brick  is  a  rectangular  solid  body. 

339.  A  cube  is  a  rectangular 
solid  bounded  by  six  equal  squares. 

A  cube  each  of  whose  edges  is 
1  inch  long  is  a  cubic  inch.  A 
cube  each  of  whose  edges  is  1  foot 
long  is  a  cubic  foot,  etc. 

340.  The  volume  of  a  solid  is 
its  contents  reckoned  in  cubic 
units. 


A  Cube. 


^ 

y 

A 

^ 

'  y  y  y 

y 

y-  y  y   y 

y 
y 

y 

To  find  the  Volume  of  a  Rectangular  Solid. 

341.  a.  If  a  rectangular  solid  is  4  inches  long  and  2  inches 
wide,  how  many  square  inches  must  its 
base  contain  ?     (Art.  329.) 

"b.  If  the  base  of  the  solid  contains 
4  times  2,  or  8  square  inches,  and  it  is 
1  inch  thick,  how  many  solid  or  cubic 
inches  must  it  contain  ?  If  it  is  2 
inches  thick,  how  many  cubic  inches 

must  it  contain  ?     If  it  is  3  inches  thick,  how  many  cubic 

inches  must  it  contain  ? 

What  numbers  have  you  multiplied  together  to  find  the  vol- 
ume of  the  rectangular  solid  4  inches  long,  2  inches  wide,  and 
3  inches  high  ? 

342.  The  volume  of  any  rectangular  solid  is  found  by 
multiplying  the  member  of  units  in  the  length  hy  the  number 
of  like  units  in  the  breadth ^  and  this  product  by  the  number 
of  like  units  in  the  thickness.  This  is  expressed,  for  brevity, 
as  multiplying  together  the  length,  breadth,  and  thickness. 


134 


MENSURATION 


343.    Oral  Exercises. 

c.  How  many  cubic  inches  in  a  rectangular  solid  6  inches 
long,  5  inches  wide  and  3  inches  thick  ?  4  inches  long,  2 
inches  wide  and  7  inches  thick  ?  12  inches  long,  12  inches 
wide  and  1  inch  thick  ? 

d.  How  many  cubic  feet  in  a  cube  3  feet  long,  3  feet  wide 
and  3  feet  thick,  or  in  1  cubic  yard  ? 

344.  Eepeat  the  following  table  : 

Cubic  Measure. 
1723  cubic  inches  =  1  cubic  foot. 
27  cubic  feet      =1  cubic  yard. 
128  cubic  feet      =  1  cord  (iised  in  measuring  wood). 

Wood  is  generally  cut  for  the  market  into  sticks  4  feet  long,  and 

laid  in  piles,  so  that 
^  p^P  the  length  of  the  sticks 

becomes  the  width  of 
the  pile.  A  pile  4  feet 
wide,  4  feet  high,  and 
8  feet  long,  contains 
1  cord. 

One  eighth  of  a 
cord  is  called  1  cord 
foot.      1    cord    foot 


Ai»J»^*^^.  ' 


contains  16  cubic  feet.      (See  illustration  above.) 
345.    Examples  for  the  Slate. 

12.  How  many  cubic  inches  will  a  box  contain  that  is  11 
inches  long,  7  inches  wide  and  6  inches  high,  inside  measure  ? 

13.  How  many  cubic  inches  in  a  beam  5  feet  long,  3  inches 
thick  and  4  inches  wide  ? 

14.  A  cubic  foot  of  water  weighs  62  J^  pounds.  What  is  the 
weight  of  water  that  a  cistern  may  contain  which  measures  on 
the  inside  4  feet  in  length,  3  feet  in  width  and  4  feet  in  depth  ? 

15.  How  many  cords  of  wood  in  a  pile  20  feet  long,  4  feet 
wide  and  4  feet  high  ? 

16.  How  many  cords  in  a  pile  30  feet  long,  4  feet  wide  and 
3i  feet  high  ? 


MENSURATION.  135 

Lumber  and  Boards. 

346.  Sawed  timber  and  boards,  when  1  inch  or  less  in 
thickness,  are  generally  reckoned  by  the  square  foot  of  sur- 
face measure.  When  more  than  1  inch  in  thickness,  they' 
are  reckoned  in  proportion  to  their  thickness.     Thus, 

2000  square  feet,  1  inch  or  less  thick  =  2000  feet,  board  measure. 
2000  square  feet,  1^  inches  thick  =  3000  feet,  board  measure.' 
2000  square  feet,  2  inches  thick  =  4O0O  feet,  board  measure. 

And  so  on. 

17.  How  many  feet  of  boards,  1  inch  thick,  in  a  tight  board 
fence,  2  rode  long  and  4  feet  high  ? 

18.  How  many  feet  board  measure,  in  a  plank  7  feet  long, 
2^-  inches  thick,  14  inches  wide  at  one  end  and  10  inches  wide 
at  the  other  ? 

Note.     First  find  the  average  width  which  equals  one  half  the  sum  of 
the  widths  at  the  ends. 

19.  How  many  feet,  board  measure,  in  a  pile  of  12  boards, 
each  board  being  J  of  an  inch  thick,  14  feet  long,  14  inches 
wide  at  one  end  and  11  inches  wide  at  the  other  ? 

PRISMS,  CYLIJSTDERS,  PYRAMIDS,  CONES,  AND  SPHERES. 

347,  To  find  the  volume  of  a 
prism  or  a  cylinder,  Mtdtijjhj  the 
area  of  the  base  hj  the  height.  Thus, 
if  the  area  of  the  base  of  a  prism 
is  15  square  inches  and  the  height 
9  inches,  a 

the    vol- 

A  Prism.  A  Cylinder. 

ume    is 
15  X  9,  or  135  cubic  inches. 

348.  A  pyramid  is  1  third  of  a 
prism  of  the  same  base  and  height, 
and  a  cone  is  1  third  of  a  cylinder  XJ^ 


of    the    same    base    and    height,    a  Pyramid,         a  Cone. 


136  MENSURATION. 

Hence,  to  find  the  volume  of  a  pyramid  or  a  cone,  Multijphj 
the  area  of  the  base  by  the  height  and  divide  the  product  by  3. 

349.  The  surface  of  a  sphere  is  equivalent 
to  four  great  circles  of  the  sphere.     Hence,  ^ 
when  the  diameter  is  given. 

To  find  the  surface  of  a  sphere,  Find  the ' 
area  of  a  great  circle  of  the  s'phere  and  multi- 
ply it  by  4.  ^  sp^®^®- 

350.  A  sphere  may  be  regarded  as  made  up  of  pyramids 
whose  bases  taken  together  form  the  surface  of  the  sphere 
and  whose  height  is  the  radius.     Hence, 

To  find  the  volume  of  a  sphere,  Multiply  the  surface  by 
the  radius  and  divide  the  loroduct  by  3. 

351.     Examples  for  the  Slate. 

20.  The  base  of  a  prism  contains  4  square  feet,  and  its 
height  is  3  feet  6  inches.  How  many  cubic  feet  does  it  con- 
tain ? 

21.  How  many  cubic  feet  of  water  will  a  circular  tub  con- 
tain, that  measures  across  the  top  and  bottom  2 J-  feet,  and 
whose  depth  is  li  feet  ? 

Note.    First  find  the  area  of  the  bottom. 

22.  I  have  a  paper  weight  made  of  flint  glass  in  the  shape 
of  a  pyramid,  measuring  4  inches  square  on  the  base  and  5 
inches  in  height.  A  cubic  inch  of  the  glass  weighs  lf|  ounces, 
what  is  the  weight  of  the  whole  pyramid  ? 

23.  How  many  cubic  feet  in  a  conical  hay-stack  that  meas- 
ures across  the  base  16  feet,  the  height  being  18  feet  ? 

24.  How  many  square  inches  are  there  in  the  surface  of  a 
foot-ball  that  is  8  inches  in  diameter  ?  How  many  cubic  inches 
does  it  contain  ? 

25.  How  many  cubic  feet  in  a  globe  2  feet  in  diameter  ? 


MISCELLANEOUS  EXAMPLES.  137 

352.    Miscellaneous  Examples  for  the  Slate. 

26.  What  will  be  the  cost  of  a  turkey  weighing  O-j^^-  pounds 
at  20/'  a  pound? 

27.  A  man  owns  240  acres  of  Western  land,  which  is  J  of  a 
township.     How  many  acres  are  there  in  the  township  ? 

28.  A  shipmaster  has  bought  f  of  J  of  a  ship  for  $3000. 
What  is  the  value  of  the  ship  at  the  same  rate  ? 

29.  After  11  §  yards  of  a  piece  of  linen  were  sold,  there  re- 
mained l^jr  yards.     How  many  yards  were  there  at  first  ? 

30.  The  old  shilling  of  New  England  was  J  of  a  dollar,  and 
the  old  shilling  of  New  York  was  J  of  a  dollar.  How  many 
New  York  shillings  equalled  3  New  England  shillings  ? 

31.  A  dealer  sold  60  bundles  of  asparagus  at  the  rate  of  3 
for  25  cents.     What  did  lie  receive  for  them  ? 

32.  By  working  10  hours  a  day  a  person  can  do  a  piece  of 
work  in  2j  days.  In  how  many  days  can  he  do  it  working  7J 
hours  a  day  ? 

33.  How  many  breadths  of  carpeting  f  of  a  yard  wide  will 
reach  across  a  room  18  feet  wide  ? 

34.  A  steamer  going  at  the  rate  of  242  miles  a  day  started 
from  New  York  on  Wednesday  at  noon.  On  the  following 
Friday  at  noon  another  steamer  going  at  the  rate  of  264  miles 
a  day  started  on  the  same  course.  How  far  apart  will  the 
steamers  be  at  the  end  of  10  days  from  Friday  noon  ?  In  how 
many  days  will  they  be  together  ? 

35.  I  bougiit  a  barrel  of  sugar  for  12/  a  pound,  and  I  can 
now  buy  the  same  kind  of  sugar  for  9 J  /  a  pound.  What  per 
cent  do  I  lose  on  the  sugar  I  have  left  ? 

36.  If  §  of  a  barrel  of  beef  lasts  a  family  3|  weeks,  how  long 
will  5  of  a  barrel  last  them  ? 

37.  A  man  put  $  576  in  a  savings  bank  October  10,  1877. 
If  it  gained  2 J  %  of  interest  every  six  months,  what  was  due 
April  10,  1878> 

38.  A  lot  of  paper  which  cost  $  748.60,  being  damaged  in 
the  store,  was  sold  at  12\  %  less  than  it  cost.  What  did  it 
sell  for? 


APPENDIX. 


Multiplication  and  Division  Tables  (Arts.  101,  137). 
1.    The  pupil  sliould  be  taught  to  write  upon  the  slate  the 
multiplication  and  division  tables.      Either  of  the  following 
forms  is  shorter  than  that  given  in  the  book,  and  may  on  that 
account  be  preferred : 

2.     Multiplication  of  2's  and  3's  (Art.  101). 


First  Form. 

Second  Form. 

2's.                 3's.  . 

2's,                      3's, 

1 

2 

2 

1 

3 

3 

2x1=:       2 

3x1=.   3 

2 

2 

4 

2 

3 

6 

2x2=   4 

3x2=   6 

3 

2 

6 

3 

3 

9 

2x3=   6 

3x3=   9 

4 

2 

8  1 

4 

3 

12 

2x4=   8 

3x4--.  12 

6 

2 '10 

6 

3 

15 

2  X  5  =  10 

3x5  =  15 

Etc 

Etc 

Etc. 

Etc. 

The  reading  in  the  first  form  should  be,  "  one  two  is  two  "  ;  "  two 
two's  are  four,"  etc.  In  the  second  it  should  be,  "  two  multiplied  by 
one  is  two'';  "  two  multiplied  by  two  is  four,"  etc.  Or  the  reading  in 
either  case  may  be,  "once  two  is  two";  "twice  two  are  four,"  etc. 

3.     Division  by  2  and  3  (Art.  137). 


First  Form. 

Second  Form. 

3.                      S. 

3.                        3. 

2 

2 

1 

3 

3 

1 

2-2  =  1 

;   3-3=1 

2 

4 

2 

3 

6 

2 

4-2  =  2 

6-3  =  2 

2 

6 

3  ' 

3 

9 

3 

6-2  =  3 

9-3  =  3 

2 

8 

4 

3 

12 

4 

84-2  =  4 

12-3  =  4 

2 

10 

5    : 

3 

15 

5 

10  -  2  =  5 

15-3  =  5 

Etc 

] 

Etc. 

Etc. 

Etc. 

The  reading  in  the  first  form  should  be,  "  two  in  two,  one";  "  two's 
in  four,  two,"  etc.  In  the  second  it  should  be,  "  two  divided  by  two 
is  one";  "four  divided  by  two  is  two,"  etc. 


APPENDIX,  139 

Solutions  (Art.  104,  140). 

4.  There  are  two  sorts  of  practice,  both  of  which  the  pupil 
needs  to  follow  at  different  times  during  his  course  in  Aritli- 
metic.  One  consists  in  giving  answers  quickly,  as  soon  as  ho 
hears  or  reads  the  problem,  without  expressing  the  reason- 
ing at  all ;  the  other  in  giving  the  analysis  or  reasoning  pro- 
cess according  to  some  approved  model.  The  former  secures 
promptness,  the  latter  accuracy.  The  teacher  will  best  decide 
at  what  time  and  to  what  extent  his  class  may  need  either 
sort  of  practice.  It  may  be  remarked,  however,  that  set  forms 
of  reasoning  should  not  be  so  rigidly  insisted  on  as  to  repress 
originality ;  for  oftentimes  the  less  formal  methods  that  come 
from  real  thinking  are  better  than  the  more  regular  processes 
that  come  from  mere  imitation. 

(Art.  140.) 

5.  The  following  form  of  solution  may  be  preferred  to  the 
one  given  in  Art.  140. 

Solution.  —  Since  2  cents  will  buy  1  top,  1  cent  will  buy  1  half  of  a 
top,  and  21  cents  will  buy  21  halves,  which  equal  10  tops  and  a  half. 
Ans.  10  tops  and  a  half  (or, 

10  tops  and  half  enough  money  to  buy  another  top). 

(Art.  117.) 

6.  The  pupil  learns  by  this  exercise  to  combine  adding 
with  multiplying  in  a  way  similar  to  that  required  in  multipli- 
cation where  there  are  carryings. 

The  following  shows  the  form  of  the  exercise  with  2  for  the 
multiplier : 

4         ^         S         7        ^  1/         6        ^         § 

^      ~i      7    7i    7^      ~s    7^    ?7     Tj 

Greatest  Common  Factor  (page  80). 

7.  We  have  seen  in  Art.  191  that  6,  the  greatest  common 
factor  of  18  and  24,  is  the  product  of  2  and  3,  the  only  prime 


140  APPENDIX, 

factors  common  to  18  and  24.  The  greatest  common  factor  of 
any  two  or  more  numbers  is  the  prodicct  of  all  the  prime  factors 
which  are  common  to  those  numbers. 

Note.     The  letters  g.  c.  f.  are  used  for  greatest  common  factor. 

8.  a.  What  is  the  greatest  common  factor  of  12, 36  and  42  ? 

WRITTEN  WORK.  The  prime  factors  of  12  are  2,  2,  and  3.     The 

12  =  2  X  2  X  3  product  of  such  of  these  as  are  common  to  36  and 

ffcf=2x3  =  6      ^^  must  be  the  g.  c.  f.  required. 

We  find  that  2  is  a  factor  of  both  36  and  42. 
We  find  that  but  one  2  is  a  factor  of  42  ;  therefore  only  one  2  is  used 
as  a  factor  of  the  g.  c.  f.  We  find  that  3  is  a  factor  of  both  36  and  42 ; 
therefore  3  is  a  factor  of  the  g.  c.  f.  Thus  the  g.  c.  f.  sought  is  2x3, 
equal  to  6. 

To  find  the  greatest  common  factor  of  two  or  more  num- 
bers :  Separate  one  of  the  numbers  into  its  prime  factors,  and 
find  the  product  of  such  of  them  as  are  common  to  the  other 
numbers. 

Least  Common  Multiple  (page  80). 

9.  As  any  number  contains  all  its  prime  factors,  a  multiple 
of  any  number  must  contain  all  the  prime  factors  of  that 
number. 

A  common  multiple  of  two  or  more  numbers  must  contain 
all  the  prime  factors  of  those  numbers,  and 

The  least  common  multiple  of  two  or  more  numbers  is  the 
least  number  which  contains  all  the  prime  factors  of  those 
numbers. 

Note.   The  letters  1.  c.  m.  are  used  for  least  common  multiple. 

10.  b.  What  is  the  least  common  multiple  of  6, 12  and  15  ? 

WRITTEN  WORK.  The  least  multiple  of  6  is  6,  which 

6  =  2x3  ^*^y  he  expressed  in  the  form  2x3. 

12  =  2  x  2  X  3  ^^^  ^^^^^  multiple  of  12  is  12,  which 

I P^  _  o     K  may  be  expressed  in  the  form  2x2x3. 

But  in  6  we  have  already  two  of  the 

1.  c.  m.  =  2  X  3  X  2  x  5  =  60      factors  (2  and  3)  of  12;  hence,  if  we  put 


APPENDIX.  141 

■with  the  prime  factors  of  6  the  remaining  factor  (2)  of  12,  we  shall 
have  2x3x2,  which  are  all  the  factors  necessary  to  produce  the 
1.  c.  m.  of  6  and  12. 

The  least  iiiultiple  of  15  is  15,  which  may  be  expressed  in  the  form 
3x5.  In  the  1.  c.  m.  of  6  and  12  we  have  one  of  the  prime  factors  (3) 
of  15  ;  hence  if  we  put  with  the  prime  factors  of  6  and  12  the  re- 
maining factor  (5)  of  15,  we  shall  have  2x3x2x5,  which  are  all 
the  prime  factors  necessary  to  produce  tlie  1.  c.  m.  of  6,  12  and  15. 

The  product  of  these  factors  is  60,  which  is  the  L  c  m.  sought. 

Note.  In  finding  the  least  common  multiple,  the  factors. of  the  given 
numbers  seldom  need  to  be  expressed,  and  the  written  work  may  be  greatly 
reduced.  Thus,  in  this  example  the  written  work  may  be  simply  1.  c.  m. 
=  2x3x2x5=  60. 

To  find  the  least  common  multiple  of  two  or  more  num- 
bers :  Take  the  pri??ie  factors  of  one  of  the  numbers  /  with 
these  take  such  prime  factors  of  each  of  the  other  numbers 
in  succession  as  are  not  contained  in  any  preceding  number, 
and  find  the  product  of  all  these  prime  factors. 

(Art.  250.) 

11.  /.  One  of  the  following  forms  of  solution  may  be  pre- 
ferred to  that  given  in  the  body  of  the  book : 

(1.)  If  she  had  put  1  knot  into  a  tassel,  she  could  have  made  4 
tassels.  If  she  had  put  \  of  a  knot  into  a  tassel,  she  could  have  made 
3  times  4,  or  12  tassels  ;  but  since  she  put  f  of  a  knot  into  a  tassel, 
she  could  make  but  \  of  12  tassels,  which  is  6  tassels.    Ans.  6  tassels. 

(2.)  She  must  have  made  as  many  tassels  as  \  is  contained  times 
in  4. 

WRITTEN  WORK.  1  Is  contained  in  4,  4  times  ;  \  is  contained  in  4 
^^g  three  times  4,  or  12  times,  and  |  is  contained  in  4 

-s—  =  6.  -^  of  12  times,  which  is  6  times.  Ans.  6  tassels. 

12.  From  either  form  of  solution  may  be  derived  the  follow- 
ing general  rule,  which  may  be  preferred  to  that  in  Art.  253. 

To  divide  by  a  fraction :  Multiply  the  dividend  by  the  de- 


142  APPENDIX. 

nominator  of  the  cUinsor,  and  divide  the  result  hy  the  numera- 
tor ;  or,  as  more  frequently  stated, 

Invert  the  divisor  and  jproceed  as  in  multiplication  of 
fractions,' 

Note.  This  rule  can  be  derived  from  the  process  given  in  Art.  250, 
thus  ; 

If,  in  the  i^lace  of  12  in  the  written  work,  we  put  the  factors  which 
formed  12,  we  shall  have  (4x3)-^2,  or  l|a,  in  which  the  expression  for  the 
divisor  is  inverted. 

13.    General  Method  of  computing  Interest  (page  123). 

The  teacher  is  advised  to  have  the  pupils  use  but  one  method  of  com- 
puting interest.  Some  teachers  may  prefer  the  following  method  to  that 
given  in  the  book. 

a.  What  is  the  interest  of  %  750  for  2  years  3  months  21 
days  at  8% ? 


WRITTEN  WORK.  The  interest  of  %  750  for  1  year  at  8  % 

SJ^O  is  g  750  X  0.08,  which  is  $  GO.    The  inter- 

est for  2  years  is  2  times  %  60,  or  §  120. 


0.08 


%  60.00  X  2  =  $  120.00  3  months  21  days  equal  111  days.     The 


37 


interest  of  %  750  for  1  year  being  %  60,  the 


*  (^0  X  ITt  interest  for  1  day  is  -j-J-q  of  §60,  and  for 

'^^     ^  -=     %  18.50  111   days  it  is  \\\  of  $60,  or  §18.50, 


1138.50  which,  added  to  §120,  makes  $138.50, 

2  the  entire  interest.  Ans.  $  138.50. 

By  this  method : 

1.  To  find  the  interest  at  any  per  cent  for  any  number  of 
years^  Midtiply  the  iirinci'pal  hy  the  rate  for  1  year^  and  that 
product  hy  the  numher  of  years. 

2.  To  find  the  interest  for  months  and  days,  Change  the 
months  to  days,  and  take  as  many  360^/^5  of  a  yearns  interest 
as  there  are  days  in  the  given  time. 

Note.  By  making  the  divisor  365  instead  of  360,  accurate  interest  can 
be  obtained. 


APPENDIX. 


143 


14.    "Weights  and  Measures. 


United  States  Money. 

=  1  cent. 
=  1  dime. 

=  1  dollar. 
=  1  eagle. 


10  mills 
10  cents 
10  dimes  ori 
100  cents       j 
10  dollars 


liiquid  Measure. 

4  gills     =  1  pint. 
2  pints    =  1  quart. 
4  quarts  =  1  gallon. 

Dry  Measure. 

2  jnnts  =  1  quart. 
8  quarts  =  1  peck. 
4  pecks  =  1  bushel. 

Avoirdupois  Weight. 

IG  ounces  =  1  pound. 
^2000  pounds  =  1  ton. 

Also  sometimes  used. 

28  pounds  =  1  quarter. 

4  quarters  =  1  hundi'cdweight. 

20  hundredweight  =  1  ton  (called  long  ton). 

Troy  Weight. 

24  grains  =  1  pennyweight. 

20  pennyweights  =  1  ounce. 
12  ounces  =  1  pound. 

As  used  in  7nixing  medicine. 

20  grains     =  1  scruple  (  3 ). 

3  scruples  =  1  dram  (  3  ). 

8  drams     =  1  ounce  (  3  ). 
12  ounces    =  1  pound. 

liong  Measure. 

12  inches  =  1  foot. 
3  feet      =  1  3'ard. 
5J  yards  or  IC^  feet      =  1  rod. 
320  rods  or  5280  feet    =  1  niilq. 


Square  Measure. 

144  square  inches  =  1  square  foot 

9  s(piare  feet  =  1  square  yard 
301  s(iuare  yards  or ")       , 

272i  square  feet  j  =  1  square  rod. 

160  square  rods  =  1  acre. 

640  acres  =  1  square  mile. 


Cubic  Measure. 

1728  cubic  inches  =  1  cubic  foot. 
27  cubic  feet  =  1  cubic  yard. 
128  cubic  feet      =  1  co]-d. 


Time. 

=  1  minute. 
=  1  hour. 
=  1  day. 
=  1  week. 


60  seconds 
60  minutes 
24  hours 
7  days 


52  weeks  1  day  or )  =  j  common  year, 

365  days       ) 

366  days         =  1  leap-year. 
100  years        =  1  century. 


Circular  or  Angular  Measure. 

60  seconds  =  1  minute. 
60  minutes  =  1  degree. 
360  degrees  =  1  circumference. 

Surveyors*  Ijong  Measure. 

7.92  inches  =  1  link. 
100  links    =  1  chain  (=  4  rods). 
80  chains  ==  1  mile. 

Surveyors'  Square  Measure. 

10000  square  links    =  1  square  chain. 
10  square  chains  =  1  acre.  [tion. 

640  acres  =  1  square  mile  orsec- 

36  sections  =  1  township. 

Numbers. 

12  units    =  1  dozen. 
12  dozen  =  1  gross. 
12  gross    =  1  great  gross. 
20  units    =  1  score. 


144 


APPENDIX. 


15.    Multiplication  Table. 


1 

1 

is  1 

2 

is  2 

3 

is  3 

4 

is  4 

2 
3 

I's 
I's 

are  2 
"  3 

2's 
2's 

are  4 
"  6 

3's 
3's 

are  6 
"  9 

4's 
4's 

are  8 
"  12 

4 

I's 

"  4 

2's 

"  8 

3's 

"  12 

4's 

"  16. 

5 

I's 

"  6 

2's 

"  10 

3's 

"  15 

4's 

"  20 

6 

rs 

"  6 

2's 

"  12 

3's 

"  18 

4's 

"  24 

7 

rs 

"  7 

2's 

"  14 

3's 

"  21 

4's 

"  28 

8 

I's 

"  8 

2's 

"  16 

3's 

"  24 

4's 

"  32 

9 

rs 

"  9 

2's 

"  18 

3's 

"  27 

4's 

"  36 

10 

I's 

"  10 

2's 

"  20 

3's 

"  30 

4's 

"  40 

11 

I's 

"  11 

2's 

"  22 

3's 

"  33 

4's 

"  44 

12 

I's 

"  12 

2's 

"  24 

3's 

"  36 

4's 

"  48 

1 

5 

is  5 

6 

is  6 

7 

is  7 

8 

is  8 

2 

5's  are  10 

6's  are  12 

7's  are  14 

8's  are  16 

3 

5's 

"  15 

6's 

"  18 

7's 

"  21 

8's 

"  24 

4 

5's 

"  20 

6's 

"  24 

7's 

"  28 

8's 

"  32 

5 

5's 

"  25 

6's 

"  30 

7's 

"  35 

8's 

"  40 

6 

5's 

"  30 

6's 

"  36 

7's 

"  42 

8's 

"  48 

7 

5's 

"  35 

6's 

"  42 

7's 

"  49 

8's 

"  56 

8 

5's 

"  40 

6's 

"  48 

7's 

"  56 

8's 

"  64 

9 

5's 

"  45 

6's 

"  54 

7's 

"  63 

8's 

"  72 

10 

5's 

"  50 

6's 

"  60 

7's 

"  70 

8's 

"  80 

11 

5's 

*•'  55 

6's 

"  66 

7's 

"  77 

8's 

"  88 

12 

5's 

"  60 

6's 

"  72 

7's 

"  84 

8's 

"  96 

1 

9 

is  9  1 

10 

is  10 

11 

is  11 

12 

is  12 

2 

9's  are  18 

lO's  are  20 

1  I's  are  22 

12'sare24 

3 

9's 

"  27 

lO's 

"  30 

ll's 

"  33 

12's 

"  36 

4 

9's 

"  36 

lO's 

"  40 

ll's 

"  44 

12's 

"  48 

5 

9's 

"  45 

lO's 

"  50 

ll's 

"  55 

12's 

"  60 

6 

9's 

"  54 

lO's 

"  60 

ll's 

"  m 

12's 

"  72 

7 

9's 

''    63 

lO's 

"  70 

ll's 

"  77 

12's 

"  84 

8 

9's 

"  72 

lO's 

"  80 

ll's 

"  88 

12's 

"  96 

9 

9's 

"  81 

lO's 

"  90 

ll's 

"  99 

12's 

"108 

10 

9's 
9's 
9's 

"  90 
"  99 
"108 

lO's 
lO's 

"100 

-ll:s 

"110 

12's 

12's 

^  12's 

"120 

11 
12 

"132 
"144 

^711711^  iIT7 


Id   I  m^ I 


'QAi 


I 


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FOB  ADVANCEB  CLASSES. 

By  WILLIAM    T.  ADAMS, 
Formerly  Principal  of  the  'Bowditch  School,  Boston. 
Intended  for  scholars  already  familiar  with  the  principles  of  pronouncing 
and  syllabication,  and  designed  to  follow  the  ordinary  spelling-book  as 
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